IIRC, the difference between 'Assault Gun' and 'Tank Destroyer' in the German armor was bureaucracy at work rather than real world application. AG was Artillery arm. TD was Army. Both were present at the corps and divisional level.
Outside of the open topped TDs (Nashorn, Marder, etc.) they were used for similar missions.
The STUG IV and Jagpanzer IV. Quite similar use of the same chasis, outside of the fact that the latter had a redesigned superstructure. Both were meant to carry the same L/70, but the artillery arm lost out to the army and stugs had to make do with the L/48
Can't completely agree or disagree with you, there are some differences still:
Assault guns provide direct fire support for infantry, destroying fortifications and bunkers, using high explosive shells. Because they are very close to the enemy, they have closed superstructures. With correct armament they can serve as tank destroyers too, but it wasn't their primary role. In game examples are StuG III E (perfect example of an early assault gun), StuG III F or G (not sure right now, which one it is - upgunned to be able to counter T-34s and also capable against infantry) and Sturmpanzer IV Brummbär wreaking havoc on infantry with its 150mm low velocity gun.
The tank destroyers on the other hand, were meant to engage armor at long distances using high velocity anti-tank rounds, so the early types were using open top superstructures believing it would provide adequate protection, because of the longer distances from the enemy (this and because some of the antitank guns just wouldn't fit into an enclosed superstructure). These were later changed to closed top, because of the dangers of flanking infantry, shrapnel and grenades. They were used mainly with Panzer and Panzergrenadier divisions, due to their higher mobility. Infantry could do with standard towed anti-tank guns. In game we so far only have the Elefant as a true tank destroyer, being superb against tanks and crap against infantry (due to previously explained different types of guns).
One last thing: an assault gun with proper armament can successfully function as a tank destroyer, while a tank destroyer cannot function as an assault gun (especially the open topped ones).