What does a "dominant" level influence over another civ culture do?

Does it give any bonuses or something else?

If your culture is Dominant over another Civilization and your chosen Ideology differs with theirs, you will be able to single-handedly cause Revolutionary Waves (~ -20 Happiness) in their empire (assuming that any other Ideology pressure modifiers have a net value of 0). In other words, "Dominant" amounts to an Ideology pressure value of 5 (the maximum) while "Influential" only produces a level of 4 (you have to factor in the pressure from other Civs as well as their pressure on you, however).

They don't give your direct bonus per se.. but Civs in Revolutionary Waves have to deal with all sort of problems; Rebellions (barbarians spawn inside their territories) and cities revolting to other Civilizations. They usually revolt to the dominant Civs but location plays a factor too.