There are three primary types of paint in use when it comes to plastic modeling; laquer, enamel, and acrylic. The primary differences, from a usability standpoint, are mainly between enamel / laquer and acrylic. I tend to lump laquer and enamel together because they have a lot of similarities from the perspective of building plastic models. They are chemically different, and laquer is more "Aggressive" than enamel, but they are similar enough in how they behave.
Acrylic has the distinct advantages of being virtually odorless and it cleans up with water. It doesn't adhere to plastic as well as enamel or laquer, and unless it is allowed to cure thoroughly painting over it can cause problems. It also dries to the touch very, VERY fast. I have a hard time brushing acrylics simply because they dry so fast; the paint will dry on the brush and whatever I'm using for a palette before I get a couple of passes done.
Enamel and laquer, on the other hand, are pretty noxious to spray. Handling the fumes can be a problem. Laquer is just plain evil. I don't use a respirator when I airbrush acrylics (although I should) but I do when I spray enamel. I avoid laquer like the plague these days, I had enough of that stuff to last me forever back when I was paint R/C car bodies. These two tend to adhere to the plastic better, and when fully cured have a much harder surface than acrylic (acrylic, after all, is a form of plastic). Painting over enamels is seldom a problem.
Personally I use mostly acrylic for airbrushing. I like being able to clean up with water or mild solvents (I usually use MikeV's solvent recipe of 2 parts distilled water, 1 part Simple Green, 1 part Windex), I like not having a basement full of paint fumes, and when airbrushing I like having the surface dry about the time I put the airbrush down. If I am going to paint over it with another color I'll usually spray a light coat of Future over it first.
If I am going to hand brush something I'll usually use enamel though. The super quick drying feature of acrylic just really gets in my way when trying to use a brush.
I'm sure other people will have more comparisons, but those are the ones that come to mind for me.