Yes, to add to what Drew said above, these 2 sentences are in different tenses. Let me explain them in the reverse order.
1) Did anyone phone me? (simple past tense) : This question simply asks about an event that did or did not occur in the past.
2) Has anyone phoned me? (present perfect tense) : This question asks about the present state. Is it true that someone has already (or "ever" as Drew points out) phoned me, or not. In fact, this is a type of "present tense". It asks for either a "0" or "1" (yes or no) answer in the present state.
Effectively, however, in conversation, native speakers would switch between the two without much thought. You can know that these tenses are indeed different when you add a time expression.
1) Did anyone phone me two hours ago? (correct)
2) Has anyone phoned me two hours ago? (not correct) -- you can not be in the present state, 2 hours ago.
*** "anybody" and "anyone" are interchangeable under most contexts, but "anyone" seems to sound better here