Do theaters do anything that amphitheaters don't (other than fit into a 2x2 space)?

If my city has perfect amphitheater coverage, do I need to worry about having poor or nonexistent theater coverage?

It's been forever since I last played, but I seem to remember theaters factoring into the Culture rating. Do excuse me if I'm mistaken.

Yes and no. It's slightly complicated. Basically each house has an entertainment rating, which is increased when walkers from entertainment buildings walk by. Point is that the entertainment requirements for houses to upgrade are only based on the rating, and not the entertainment buildings that supplied the rating. So for instance you could build Colosseums instead of theaters or ampitheaters, and your houses would still upgrade as far as, say, Large Insulae (haven't checked).

I don't know the exact figures, but from what I understand it's something along the line of:

walkers from a
- theater add an entertainment rating of 10 (15 is there is a show, ie actor's colony)
- amphitheater add an entertainment rating of 15 (20 if there's two shows, ie actor's colony and gladiator school)
- colosseum add an entertainment rating of 20 (25 if there's two shows, ie gladiator school and lion pit)
- hippodrome add an entertainment rating of 20

The walkers can therefore increase the entertainment rating of a house to a maximum of 80 out of a 100. The remaining 20 is determined by city wide coverage of your entertainment buildings. That is, if you have perfect theater, amphitheater, colosseum and hippodrome coverage, then all of your houses will have at least 20 entertainment rating.

So why would you want to build theaters? Well, on levels where you can't build a colosseum you might not be able to increase the entertainment rating of houses high enough without amphitheaters and theaters. That is, your houses won't be able to advance far enough without amphitheaters and theaters.
Or, like lanipcga said, to increase the culture rating. A large part of the culture rating is determined by the coverage of your entertainment buildings. That is, a city with perfect amphitheater coverage will have a smaller culture rating than the same city with perfect amphitheater and theater coverage.

Oh yes one last thing that took me a while to realize. The coverage of an entertainment building is not determined by the proportion of houses that have access to the entertainment building, at least not mainly. It's dependent on how many entertainment buildings you have versus your city population. For example, a theater only services 200 people even though it's walker may spread entertainment to houses with a combined population way way more than 200. For the purposes of increasing your culture rating due to a specific entertainment building you'll have to make sure you have enough of those buildings. Even if they are somewhere outside the city in a little shack village :P

So I gather that if I have perfect amphitheater and colosseum coverage (and both types of entertainment for both buildings) and some dwellings still have "some entertainment, but not enough", theaters are my best bet for rectifying that.

If by coverage you mean that the houses have access to theaters and amphitheaters as indicated by their respective overlays, then yes. But if you mean coverage in the sense of city coverage on the entertainment advisor's screen, then no, not necessarily. From what I understand it's possible to have perfect city coverage, while at the same time having houses with no access to the entertainment building.