I believe your teacher is incorrect. Russia is not the same as the USSR as Russia is one republic in the Union of Soviet Socialist Republics. Perhaps that is what your teacher meant. Unless there's some obscure nuance I'm unaware of the USSR and the Soviet Union are the same thing.

Soviet Union is just a shortened version of Union of Soviet Socialist Republics, so your teacher is incorrect. I'm not sure where they got that idea from. Try asking what he thinks the difference is. I'd be curious to find out the origin of this misconception.