Is there any purpose to culture at all, other than for a cultural win or histograph points? (I hate histograph wins, but I'll take them.)
To understand Culture, you have to understand where it came from, why it was designed as a mechanic in the first place.
In Civilisation 1 and Civilisation 2, each town had the normal cross radius of interest, the area within which it's workers could work squares. You put down a city and that was your border. There was no Culture expansions which moved your borders ever further into the ether.
The annoying aspect of this system, and one which raised a lot of annoyed voices, was instances where your territory juts out a little bit, you know, those weird jagged edges and pointy bits going out to sea. No matter how well you positioned your cities it was just a complete pain making sure
your bit of land was fully controlled by your borders. If you left even one square un-bordered then the AI would drop in and put up a city.
So in Civ 3, firstly, I do believe the AI is programmed not to place a city directly onto your border and never inside your border. Corner to corner maybe, and maybe they will right on a border, but it's not the usual, they will normally only settle your border +1 square away. Having borders that expand also gets rid of those awkward squares that no amount of tessellation will allow you to cover them with the regular cross shape.
These expanding borders were called Culture. And they have many uses, but you wont ever win a game (other than culture victory) by waiting for your Culture to subsume the rest of the planet. It's mainly just a neatening tool, a way to more accurately allow a player to fully dominate a region they are obviously dominant in.
The resultant positive aspects or general game uses of Culture are as follows:
Barbarians will only build camps in the fog. Your border clears the fog by +1 square each time they expand. So, if you have an awkward Mountain range that sticks out and is going to always be in fog (because you can't settle on a Mountain), instead of stationing a soldier there permanently to push out the fog, you're nearby city can eventually cover that area with border expansions, freeing up your soldier.
Sometimes resources are right in the middle of you and another Civ, border resources. You can have Culture competition to dominate the resource (or just favourable squares).
Sometimes, a close-by neighbouring Civ's city will be near one of yours and the Culture war will go beyond lines in the sand and become a full Cultural Conversion. This is actually quite rare and often happens to towns with no culture at all or ones with low Culture that are surrounded on many borders by one with high Culture. My own experiences of Cultural conversions extend to:
Ai Civ capital on neighbouring Island, the borders touching one of my coastal cities with no Culture, I lost my city to a cultural conversion.
Ai invades my lands while I'm going for a Culture Win. They take one of my too-many-to-defend cities in the middle of a huge island. The extent of the radius of that city is one square, because it has zero Culture and all the close-knit surrounding Culture borders push it in from every side. After 20 turns of it not naturally flipping back to my control I quit the game in disgust, on principle.
Sometimes the Ai will forward settle, sending out a city right next to your borders when the rest of their civilisation is miles away. If you quickly build Culture around it you can normally win it over with a Cultural conversion without having to war it out the way.
When you invade a Civ, each turn their cities make a roll for "flip-back" to the Civ you're attacking based, I believe, on Culture rating. Most human players quit if they think they're being overrun, so rarely see it on their cities, but if you experience enough situations where you lose cities but stay in the game then you'll probably see it happen with your cities.
The more unknown quantities of Culture involve how they effect international relations and the extent to which an enemy Civ goes into resistance when you take their cities. These are the kind of topics where the stat junkies might get really interested. For example, I often have Ais go to war with me right after a telling border expansion, while I also might have better trading terms with a Civ that is in awe of my culture. I might invade one city and get 6 resisters, but invade it's neighbour and get only 2 resisters, even though both had populations of 10.
If any opposing unit wanders into your borders during peacetime you get a choice of options as to what to tell them, either "please can you move them" or "either move them or declare war". You really want both of them to always be options, but most of the time it just gives you "please move them" which is really frustrating if you really want them moved or you really want them to declare war on you.
Culture points play a large part in expanding your private palace, if such a thing interests you.
So it's something that's not crucial to a non-Culture win, but something that has it's part to play in wider mechanics and general gameplay enjoyment, together with a few, very scenario dependent areas where it's crucial, or at least very important.