While both are tubeframes with composite bodywork (the C5-R tubeframe is built inside a the production car which is mostly a tubframe but has parts similar to a unibody), the main difference is in the drivetrain and suspension.
The Trans-Am Corvette has a transmission mounted to the engine via a bellhousing with a live rear axle - that is a solid axle where the differential is sprung. The C5-R has a transaxle at the rear, where the transmission and differential are one piece and bolted to the chassis; it has independent suspension and only the wheels/tires/brakes themselves are sprung. The C5-R has a carbon clutch, while the Trans-Am car doesn't.
According to Grand-Am GTS rules, both can use the LS6 motor in 6-liter form, though the C5-R has to run it with restrictors, while the Trans-Am car has to run it with stock intake manifolds, and a rev limit.
Minimum weight is 2400lbs for the C5-R and anywhere from 2400 to 2600lbs for the Trans-Am car depending on what engine is used.
The Trans-Am car is a bit wider by a few inches as well.
