Umm that tune on Gameboy is Korobeiniki.
https://www.youtube.com/watch?v=NmCCQxVBfyM I just double checked that one with a youtube video of the Tetris audio. (Oddly also came up with the verbalized version with the old Soviet choir which is impressive.)
https://www.youtube.com/watch?v=W80xBX0jK-c&feature=kp
There are plenty more differences in the game too. NintendoAge went over this since Thor posts there and also the second runner up to the old NWC, both Tetris champs in their own right. The SNES game has different timing, scoring and other requirements as well and I really just don't remember it all.