firstly, Illustrator cant handle animation, it's just for drawing.
The best thing to tell you to sum up both packages is that flash is best used for cartoon graphics and animation, but if you really want to start looking into realistic vectors and complex illustrations etc. then your best bet is to look into illustrator.
Tool differences:
The method you describe in flash, you can actually do in illustrator aswell, but I prefer to use flash for it as illustrator is... more precise (sound like a good thing but sometimes it's not)... and isn't as easy to use (that said, there is some variety in how you can distort the shape in illustrator, which is a plus).
However, illustrator has the pencil tool, which is the quickest way to make accurate vectors, period. Much more accurate and faster than flash.
Of course to make accurate vectors you could use the pen tool in flash, or in illustrator, or in photoshop. But generally speaking it's actually much better in photoshop so i would stick with that
Although Flash's autotrace is pretty poor, illustrator has a feature called live trace which is surprisingly good. This feature is only availible on creative suite though.
Features:
In illustrator you have countless features which you can apply to your vectors (like blends, which save you a lot of work), which really saves your ass if you need a quick texture or the like. I used to do vectors in flash but I hated the size restrictions and the clumsy tools.
Of course in flash you don't really have any features acossiated with your vectors (not particularly time saving or usefull ones anyways)
Downside:
The downside is of illustrator is that it takes a lot of learning and using to get used to the interface, and with the way opacity masks work, I wouldn't bother doing masks in illustrator, do them in photoshop or equivalent. It's worth learning it though if you want to make your illustrations better....
Ok that's about all I can think of, but I use illustrator daily so if you have any questions just post em here and ill do the best I can to answer