The biggest difference between a classical guitar and what everyone considers a typical acoustic is the strings; a classical uses nylon strings while an acoustic has steel strings more similar to electric guitars. Steel strings acoustics are by far more popular, but it's a preference thing. Each have their own individual characteristics.

The nylon strings not only give a "warmer" tone than the "brighter" steel strings, but they're easier to fret as well.

Another difference is that classical guitars have wider fret-boards that separate the strings a bit more; this is supposed to ease the fretting of cramped multi-finger chord shapes, scale-runs, and facilitate the finger-picking techniques of classical and flamenco.