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[drc]

A visiting professor is often paid by his main university, but also may have grant money and may be paid by the university he's visiting. Some combination of the three pays the way, varying from case to case. Visiting professors or scholars can be on paid sabbatical from their home university or unpaid sabbatical, the latter usually meaning they have to get funding from somewhere else. If he/she teaches a course the university he is visiting would probably be required by its own rules to pay him for this service.

A visiting assistant professor is a temporary employee, probably paid by the university he is teaching at. He is hired on a temporary basis, perhaps a year, to teach some courses. As a temporary, ephemeral employee, he doesn't get the perks and privileges of long term hires. A visiting assistant professor can be there to fill in gaps or perhaps to pick up the slack for professor who is on sabbatical or ill.

The visiting designation is like with an American visitor on vacation to Paris. France may welcome you and may even pay your way (contest winner?), but you're just a visitor, have to go back home and your trip and visa in no way allow you to claim to be a citizen or resident.