Simply put, when you capture a city in Civ5 you have the choice to "Annex" it, turn it into a "Puppet" state, or just "Raze" it. Annexing basically turns it into a regular city under your control, however every annexed city adds five unhappiness to your empire until you build a courthouse there. Although i haven't actually tried it yet myself i'm told that courthouses take a long time to build, can't be rushed, and have a fairly large upkeep cost. If you raze a city it disappears (after a couple turns) and you can resettle the spot with one of your own settlers, or just leave the area empty. However if you turn the city into a puppet state you "own" the city, but you have no control over it. The AI will choose what to build for you, but you get the gold, science and culture it produces, _plus_ you don't incur any unhappiness from it. The only downside is that the AI tends to focus on military improvements and yet never builds any units. This means that every puppet city ends up with a ton of useless buildings that drain the gold the city would otherwise be providing. Depending on what techs you have available and what social policies you've chosen the gold net worth of the city may approach or even surpass zero, though i think that at least in my first game i never reached that point. So when you first puppet a city you get an initial influx of gold that, on average, slowly decreases over time. However you still get all the science and culture, so it's still definitely a net positive.
So all in all it might seem like the only difference would be whether you wanted more gold (in the long run) or not. However there's on other key difference. One of the big new things in Civ 5 is the Social Policy system. You accumulate culture points like in past games, but now you spend them on what is effectively a leveling system for your civilization. There are ten different social policy tracks, but you don't level up in each of them individually. When you hit the next "level" you can choose which policy to spend the point on. In a lot of ways it's like the skill trees in Diablo 2, and like in Diablo 2 (if instead of branches within each class you were able to choose between all the classes' trees at any time.) And like Diablo 2 since each level costs more than last and you tend to get better bonuses later on in the tree, you're much better off focusing on one branch at a time. Unfortunately(?) each tree is fairly small. Every policy has exactly five items inside it (although just opening up the policy to begin with requires one level and usually provides a smallish bonus) which means the individual trees are neither very deep nor very complex.
The catch is that the cost is scaled not only by how many levels you've already obtained, but by how many cities you control. "Control" being the key word, because puppet states don't count towards that total! Since a cultural victory requires that you max out five different trees, the best (and same say only) way to win by cultural victory is to build a couple cities on your own, then conquer and puppet state as many other cities as you can, so you obtain their culture without increasing the cost of social policies. I was rather inefficient about choosing my policies, but since i only built four cities and yet owned another ten or so puppet states i was still able to pull off a cultural victory. I actually opened up two trees that i didn't finish, with three items chosen between them, for a total of five "wasted" picks. Which means that with the five completed trees i made a total of 35 social policy picks/levels. Towards the end levels were costing about 3000-4000 culture each but i was gaining 300-400 culture a turn.
So the short version is: if you want to win by culture (or even just gain the immediate benefit of a lot of different policies) then found 2-4 cities of your own and conquer everything else you need. Firaxis is planning a patch that will actually fix a lot of the problems about dealing with puppet states without nerfing them in any way that i can tell, so for the moment at least it looks like that strategy is only going to get more powerful.
For my second game i decided to try out a strategy i read by a guy who's done several recaps of Civ 5 games so far, and apparently did some QA work for Civ 4. The basic setup was called "Always War," and involved a Pangaea continent on "King" difficulty, playing as the Aztecs and declaring war on every AI as soon as you meet them.
In theory it was also supposed to be with no city-states, but i forgot to adjust that setting, so i just ignored them. It was a little rough going at first (and involved a restart or two,) but i eventually worked up to about 8 units, 2 Jaguar Knights, 2 Horsemen, 3 Archers and 1 Catapult and start steamrolling the AI one city at a time. By the end of the game i'd upgraded to Swordsmen, Knights, Crossbowmen and a Trebuchet, all anywhere from level 4 to level 8. I'm not sure, but i think leveling up units is rather harder than it used to be. Perhaps that's because it's been made so easy and effective to upgrade units that you're expected to have them around a long time. Still, it seems like using the same units for constant warfare from when they were first built in the BC area until the end of the game in the 1600s (i think) should have resulted in higher levels than that.
So i didn't do as well as the guy i got the idea for the challenge from, but the it really wasn't that hard once i figured out the right tactics to get the AI to send all its units out to attack on open ground where i could slaughter them easily. Dealing with one unit per tile can be difficult at times, but the AI seems to have a harder problem with it than humans, and it's definitely not so good at coordinating its attacks.
I'm not sure what i'm going to try next. I may actually play as the Egyptians again, but on the easiest setting this time (there seems to be a Steam Achievement for every difficulty setting =) and see if i can manage to build every single wonder. It's a fun little challenge that's only possible (for me at least) on the very easiest difficulty settings.
Edit: Checking some exact numbers now that i'm home. Apparently i misremembered my culture at the end of the first game. The save right before the last turn the next social policy cost 5385 culture, and i was getting 859 culture a turn. As opposed to my second game, where the last turn i had 605 culture and was making 37 a turn.
1st game: Finished 1952 AD (Culture) 372 turns, 4200 points
2nd game: Finished 1635 AD (Domination) 237 turns, 1782 points