Tell me, what is difference between Stug 3G and tank destroyer? Is there any? Just becouse it is classified as assault gun does not mean it is assault gun anymore. That is my point. If you want to say Stug 3 G is assault gun, please tell me why is it assault gun and not tank destroyer. Differences are what?
As gundammerc already pointed out, it depended on the formation it was assigned to.
Looking at design differences though, the gun used is the biggest difference (taking the StuG IIIG, the JgdPz IV).
The StuG IIIG (and F/8) series had the 7.5cm StuK 40 L/48 gun, this gun was performance wise similar to the Pz IV G and H's gun.
The JgdPz IV/70 (most produced variant being the L/70) carried the Pak 42 L/70 gun, which performance wise was more similar to the panthers gun.
First difference you will notice is classification difference, this is because a gun assigned to the StuG was not considered an anti tank gun, it was considered a Sturmpanzerkanone, which means storm tank cannon, while the gun the jagdpanzer had was considered a panzerabwehrkanone which means tank deterrence cannon.
Performance wise the the L/70 generally outperformed the L/48 in almost all anti tank respects.
Along with this a tank means nothing without the doctrine around it, and the doctrine surrounding allocation and deployment of StuGs generally differed from that of the jagdpanzer, with the StuG being intended to be deployed in well StuG battalions, or in other words assault gun battalions, while the jagdpanzers were to be assigned to well Pz jäger battalions (yes these names are really original aren't they?) both were more often then not assigned to tank divisions later on in the war though.