Driver's suits are typically one-piece jumpsuits, whereas crew fire suits are typically two- or three-piece (pants and jacket or pants, vest, and sweatshirt with fireproof sleeves).
Driver's suits are more protective. If rescue folks have to drag you from a burning car by your arms or shoulders and you are wearing a crew fire suit, there is a good chance that your midriff area will get exposed and likely burnt. In a one-piece suit, that can't happen.
That said, both types are legal for Lemons. Jay has, in the past, recommended (although not required) that drivers wearing multi-piece firesuits also wear fireproof undergarments (even on multilayer/5 rated suits) to reduce the chance of burning one's belly.
FWIW, I proudly wear a horrifically ugly multi-piece crew suit and I love it. Where else in the racing world can you wear a Heathcliff Kitty Litter-sponsored racing suit and not get laughed out of the paddock?
Pat Mulry, TARP Racing #67
Mandatory disclaimer: all opinions expressed are mine alone & not those of 24HOL, its mgmt, sponsors, etc.