The 'big fat" lips are generally not from Botox, they tend to be from 'Lip Fillers'. It's a gel-type substance which forms like a temporary implant in the lips to make them puff out - Products which are injected into the lips to plump them up, make them big and juicy..I agree sometimes way too much! On the other hand Botox is an injection which paralizes part of the muscles, not making them bigger.
Recently Botox has been used on lips (and I tried it), it doesn't make much of an obvious difference - it's generally more used to stop wrinkles. Some people have gone too far with this product too. At least it wears off within a few months!