First, my big irritation with designers who hand over work they never viewed in 16 bit -- the color shifts can be really rugged. Colors in foreground images and background images may not match on many browsers, and hex defined bgcolor is often a mismatch as well.
Second, for most practical purposes 24 bit and 32 bit color depth are the same right now. It's just that 32 bit includes an 8 bit alpha channel, useful for designating degrees of transparency (in PNG images, for example). However, major the browsers still aren't up to supporting this very well, as far as I know.