Wheel-lift towing lifts one axle of the vehicle (usually the front) while the other rolls. Cheaper, faster to hook up. Fine for front- or rear-wheel-drive vehicles with the driven wheels on the lift.
Flatbed towing loads the entire vehicle onto a tilt-bed truck. Slower to hook up but no wheels touch the road. Required for AWD/4WD vehicles, low-clearance vehicles, vehicles with damaged wheels or suspension, and vehicles being towed for long distances.
AWD vehicles wheel-lifted on the wrong axle can damage the transfer case to the tune of $2,000–$5,000. Some AWD systems are more tolerant than others; some are extremely sensitive. Don't take the risk.
Low-clearance sports cars (Corvettes, Lambos, lowered Civics) need flatbed approach ramps to avoid scraping the front splitter or air dam. Standard wheel-lift hookups damage the front fascia.
If you have any doubt, request flatbed. The slightly higher cost is far less than potential damage.