You can search for the difference between "perfect" and "imperfect" aspect of past tense. The concepts are quite clear, the only problem is that English is such a mishmash language that it doesn't act consistently. But you'll find that the distinction is applied clearly in Spanish.
Anybody doesn't sound right in English. As a native English speaker, I can't really tell you why. I just know it sounds very odd. I would say that somebody is more of a "searching for a specific person/group" word and anybody is a "free-for-all/everyone is invited." I'm not sure if this makes sense to you. It's hard to explain. Hopefully someone else will come along to explain it better than me.
If I say, "somebody (or someone - they are equivalent) doesn't believe us," I mean there is a person or persons who don't believe us. If I say "Not anybody believes us," I mean that no one, or nobody, believes us. And we simply would not say "Anybody doesn't believe us."