I've never heard anyone use adulate and is not a vocabulary word worth remembering. Depending on what you want to say, there are other words more useful: to praise, compliment, suck up, fawn over, express admiration for...

The most important difference is that the word "flatter" is common while "adulate" is MUCH less frequently used! So if you are talking, it is better to use flatter. However, there is another difference if you want to know: flattery is often considered insincere while adulation may or may not be.