Backgrounds in general serve a dual purpose in character generation. The first is to provide a list of starting equipment very quickly. Their second and more important function is to act as an aid to roleplaying, by providing just enough in the way of characterization, backstory, and purpose to make THIS fighter you're playing today different from THAT fighter you played last week (you know, the one who DIED and left you in need of a slightly different character concept for next session.) The most important difference between the starting equipment packages in the original version of Into the Odd and the starting occupations from I2TO Bastionland is that the occupations don't just tell you WHAT you're carrying, they tell you WHO is carrying those things.
In DCC you just get a job title, a weapon, and one item; in I2TO you get a bit more equipment, a sentence of describing your career, and a you roll on a pair of tables that give you a couple more sentences of backstory along with a couple skills or special abilities or what have you.
In 5e, your background gives you a couple skill proficiencies, a tool proficiency, a language or two, and a package of starting equipment. (You'll get more of each of these from your character's race and class as well.) They give you tables to roll if you want Character Traits, Ideals, Bonds, or Flaws. (Which you might, since those can be sources of roleplaying insight like I was just praising other backgrounds for.) They give you a "Feature," which is a special ability that typically relates to downtime activity.
They also almost all invite you to co-create part of the campaign setting with your DM.
I didn't notice that last one previously, because I didn't read any of the backgrounds closely enough, but that's kind of a big deal. If you follow the 5e's instructions to let each player co-create part of their shared campaign world as part of character generation, you probably end up with a "Session 0" of the type advocated by Dungeon World, Freebooters on the Frontier, and Beyond the Wall, among others. If you follow those instructions, then it seems that instead of the DM developing a campaign setting before play starts, the beginning of play is actually the DM and the players creating elements of the world together in an imaginative collaboration that happens at the same time as character generation.
Of course, nothing forces you to follow the instructions as written. If you're a DM who does have a campaign setting already, then you could simply tell the player "choose from this list of gods" or "choose from this list of military organizations" rather than asking them to invent one with you. In fact, you could pretty much automate the choices. 5e does a little of this already. It doesn't ask the player to invent an artform for their entertainer or a criminal enterprise for their charlatan, it simply asks them to roll on a random list to discover which one is their specialty. So you could simply take that process further and randomize all the choices 5e asks players to make about the world. If you have a campaign setting already, this would make char-gen that much faster, let you skip Session 0, and start playing the game that much faster. Over at Tales of the Grotesque & Dungeoneque, Jack is doing just that with his Cinderheim setting. (Clever Jack also noticed that there are 12 backgrounds and 12 character classes listed in the 5e Player's Handbook, so you can REALLY randomize character creation - and add a bit of I2TO-style random backstory - by making tables for EVERYTHING.)
You could also do the opposite and expand Session 0 to include even more worldbuilding tied to the character backgrounds. (One again, Jack got their first on this one.)
So, as I said, that's one thing that 5e backgrounds provide that's unusual, at least compared to DCC and I2TO. (Not unusual, obviously, for Dungeon World or other games that emphasize co-creation.) The other thing that's a little unusual is the Feature, which provides an ability that you'll probably use during downtime. My initial impression when I skimmed the backgrounds was that they all just had a Feature that entitled you to free room and board somewhere in the game world, but on closer inspection, there is some more variety, though not so much that every background is unique. I'm going to go out on a limb and guess that Background Features must have seemed challenging to design to WotC's staff writers, because several of them are basically repeated in the Player's Handbook, and because when they suggest additional backgrounds in other places, they typically suggest just borrowing one of the current Background Features rather than supplying a new one.
When I was running my Island of the Blue Giants campaign (I'll pick it back up again someday, I swear), and using Zenopus Archives's OD&D backgrounds for human characters, I had this idea that the alien weirdos on the island wouldn't do any of the normal NPC stuff, and the only way to get those tasks accomplished would be to have the right background, or be forced to do without. So if you had an innkeeper in the party, you could convince people to let you pay them to sleep inside their buildings, but otherwise, you'd have to camp on the outskirts of every village you went to.
I relate this anecdote, because it connects to 5e's Background Features, and to the oft-repeated old-school lament that if you give one character the explicit ability to do something, you're implying that characters NEED the ability in order to do the thing. The go-to example is that giving thieves the ability to pick locks and do badass ambushes, you're implying that other characters can't do that. This doesn't have to be a lament though - remember my idea to deliberately award abilities to certain characters because I don't want others to have them. But with that in mind, if you're going to write a Background Feature that says a character can do something, then it should be clear how that task will go for characters without the Feature. And judged on that criteria, I think some of the 5e Features could be better written. There's a lot to criticize about the writing of feats in 3.5 and Pathfinder, but I will say this at least, if you take a feat that basically says "you punch real good" then there's also reminder text to make sure you know know what normal punches are like. With 5e's Background Features, it's not always clear.
To see what I mean, let's look at the sailor's Feature, "Ship's Passage." This ability lets you ride for free on any ship you want, but the DM has official permission
A second interpretation is that other characters have to pay cash to ride on ships, while sailor ride free but get sidetracked by adventure. Except honestly, that sounds worse than just settling your check. I can see that a player choosing the sailing background might mean that they're interested in sailing adventures, but what I just described feels less like giving them the adventure they want and more like penalizing them for picking that background by throwing extra obstacles in their path.
But solving that problem would require the text of the Feature to be different than it currently is. I want it to be something like this - most characters are landubbers, they have to choose between paying cash or getting taken for a spin on an obligatory nautical adventure, but if you're a sailor, then you my friend, you ride for free AND get to arrive on time. I'm a little bit nervous about suggesting an ability that basically says "use this ability to skip being forced to play D&D", but on the other hand, if EVERYTHING'S a hassle, then your highly-motivated, goal-oriented characters might never actually get anything done, because every time they take a step down the path, two new paving stones spring up between them and their goal. Maybe sometimes what you need is an ability that says "skip THIS hassle and move on to the hassle you're actually interested in." It's the same argument in favor of character skills. We can waste an hour of real time narratively describing searching a room in order to find nothing, or we can roll the dice to see that we found nothing, and get on with having a game-night where we visit more than two rooms of the dungeon before everyone has to go home. In one sense, a quest is nothing but a series of hassles that stand between the player characters and their goal, but if the solution to EVERY problem just creates more troubles that get in the way, then no one is ever getting to Solla-Sollew. You need some decisive solutions, or you just get infinite regress. You go on a quest hoping that you'll complete it, not that you'll just be led further and further afield forever. All of which goes to say, that maybe one possible use for Background Features IS to skip specific narrative hassles so you can just move on to the next part of the game.
Another possibility would be to use Features to award some kind of narrative boon. One type of narrative boon could be relationships with NPCs. So, play a noble, get a loyal servant; play an urchin, get a fellow-orphan who tags along on all your journeys. I suspect there's more opportunities for this sort of thing, although for now, gaining access to special NPC followers that you can't get any other way stands out as an appealing option. (Like the way that ex-law enforcement characters on tv shows always have "a buddy on the force" so they can call in a favor to pull a copy of the criminal record of anyone they want.) Whatever other benefit you offered, it would need to be something that money can't buy, or at least, it needs to be expensive enough that it wouldn't otherwise be available to low-level characters. So maybe the sailors OWNS a ship, or at least has more-or-less permanent, unlimited access to one for the purpose of adventure - whereas other characters CAN'T just pay cash to ride on a boat because it costs a king's ransom. They'd have to form a relationship with an NPC who could offer them passage as a favor, while the sailor skips that step. Because they already have that relationship, they can just waltz right up onto the boat. (Is THAT what the "Ship's Passage" feature is supposed to mean?)
In the 5e Dungeon Master's Guide, they hold up the sage's Feature, "Researcher," as an ideal to strive for. The feature doesn't allow the sage to know information, or give them a bonus on rolls to know things; instead it says that if they don't know the information, then they will know where to go to find out. So, according to the DMG, this is a Feature that provides an impetus to go on quests, by providing the players with goals to quest for. (Although again, I wonder about what's implied about how this goes for non-sage characters? Wouldn't they just go hire an NPC sage, who'll either tell them the information, or tell them where to go find it? Is that even actually more difficult? Is the ONLY hassle you're skipping the hassle of hiring one NPC who wants you to hire them?) I'm not sure many of the other Features accomplish the stated goal of providing adventure-seeds. I suppose if the noble owns a building that occasionally has problems you have to go fix, or if the acolyte's temple needs you to run errands for it from time to time, that might rise to the level of providing quest-fodder. But as I said before, I would want it to feel like you're rewarding each player for their choice of character background, and these ideas feel more like punishments. No one wants to take a week off from their Quest for the Holy Grail because Camelot needs you to mow the lawn again.
The Features I mentioned earlier, the ones that give you room and board somewhere, actually do meet my standard for removing hassles, even if the hassle they remove isn't one that seems super relevant to me personally. So far when I've played 5e, I've played it as episodic and mission-based. I haven't known or cared much about what my character got up to between sessions. If I'm already not imagining where she sleeps, or deducting lodging fees from her treasure count, then the chance to let her sleep somewhere for free isn't especially enticing. Just because it doesn't interest me, though, doesn't mean there's anything wrong with it. In fact, if that's the case, maybe it's good if all the Features are equally useless to the way I game. That probably just means they're all serving a function in someone else's play-style. However, it also means that I want to keep brainstorming to think of features that might all be equally use-FUL to me.
Some of these features may also be sneaking in more co-creation below the radar of my attention. The sailor, who as I mentioned, gets free passage on a boat. Well, what if it's a specific boat? It probably has a name, a little backstory, and an associated NPC or two. Those all need to be made up, presumably (if we follow the lead of how 5e asks us to make up other things) by the sailor's player and the DM working together collaboratively. The same goes for the acolyte and the entertainer, who get free room and board somewhere. My initial read on those was just that, in whatever village the characters end up in, the bard or acolyte will be able to get a free place to sleep for everyone. (That's more or less how it goes in season 3 of Robotech, where Yellow Dancer can put on a concert in every town her team passes through.) But maybe if the characters have a stable home base instead of being on the road, these Features should involve co-creating ONE temple or A theater that the characters have specific connections to.
In a way, some of these Features feel like they're a way to decide which specific take on a general activity will occur in the campaign. So for example, your characters are going to have to sleep somewhere, but drawing on the Background Features helps decide if you're being hosted at a temple, camping out in the dressing rooms of a theater, or what have you. It's not like if you DON'T have those backgrounds then you won't have anywhere to sleep, it's just that if you DO, then what could be a fairly generic activity becomes slightly personalized. I2TO does something like this too. In a Bastionland campaign, every adventuring party starts off owing a giant debt to somebody. But precisely WHO you owe all that money to depends on which character background you use to decide on the nature of the debt.
One final use for Backgrounds occurs to me, and that's as a limitation on disguises in any sort of intrigue game. In Person of Interest, the character of Finch is a computer programmer who becomes a billionaire. When he goes somewhere in disguise using a false identity, he can portray a tech worker or a rich man, BUT he can't formulate any plan that depends on him passing himself off as a tough guy. Reese is a former soldier who got trained as a tuxedo-wearing spy and assassin. Other military personnel recognize him as a veteran, hit-men instinctively know he's a killer, and and he wears a suit well enough to convince businessmen that he's only mercenary in a metaphorical sense, BUT no one would ever believe he was a social worker or a nurse. So maybe in a 5e game that involves disguises and spying, the players HAVE to come up with plans that rely on their characters pretending to be the same kind-of-person as their backgrounds, even for their fake identities. The sailor can pretend to be a cabin boy or an admiral, as the situation requires, but they won't fool anyone if they try claiming to be an entertainer. Their Backgrounds provide some scaffolding for the players' creativity in coming up with their cunning plans, and maybe lend a little continuity to the characters by ensuring that their successive fake identities all have SOMETHING in common. And like I just suggested, if you're playing an intrigue game, there ARE going to be disguises, of one sort or another. Drawing on the Backgrounds just helps to personalize that mandatory activity based on decisions the players have already made about what they want their characters to be like.
The thing that got me thinking about this is that if I'm able to run anything like the Nutcracker Princess campaign, then room and board is going to be provided to all characters as a matter of course. The hospitality of the local monarchs is NOT to be questioned. So any Feature that just says "you get free room and board" is de facto useless in that setting. And as I said before, I'm sort of okay with it if ALL the Features are equally useless, but I'd be even happier if I could think of replacement features that would be useful in that game.