"Have you invited him" feels like there's an expectation for you to invite him. "Did you invite him" is more asking about the state of him being invited. I will say though, that depending on your tone of voice, "Did you invite him" could also imply some sort of expectation.

The first one is a present perfect form, and used for present and something it has effects in present, The second one is a simple past form, and used for something started and ended in the past, Though you can take a look on grammar websites.

Have you been to the new restaurant? Have always liked to learn new languages? Have you invited him in the past? Have you invited him to you home? Have seems to have more to do with time and prior episodes of inviting. Did you invite? This is more of yes/no question.