None. It affects the AICivs' attitude to the human player (and presumably also to each other, not that you'd notice!).
Basically, if you're running e.g. a Republic, the AIs who favour Republic (such as the Romans) will be (marginally) friendlier to you (not sure if they might give you better trade-deals as well?), but the AIs who shun Republics will be less- or un-friendly. Etc. for the other combinations.
Of course, if you've had any wars with the Romans (and even if they started it), they'll hate you regardless of what gov-type you're running. So the moral of the story is, don't worry about it, because you can't make friends with the AIs anyway...