There are actually two versions of Tetris for the Game Boy - v1.0 and v1.1, each having only the slightest change in music.
In v1.0, Music A is not the Korobeiniki we all know and love, but an unknown title that more closely resembles the Tetris music on the NES. After only a few thousand copies of v1.0 were made, Music A was swapped in v1.1 to the classic Tetris tune. This is the only difference between the two games.
If you start on levels 1-9 instead of level 0, it levels up differently. From level 1 it goes to level 2 at 10 lines instead of waiting till you get to 20 lines. From level 9 it goes to level 10 at 20 lines instead of 100 and then goes to level 11 at 40 lines instead of 110. There also seem to be minor differences with sound effects, like you can hear rotation/tapping sounds if you try to move your next piece during a non-Tetris lineclear, and the level-up sound is barely anything.