While both are Tetris, TGM is very different, and closer to modern Tetris than the NES version.
The NES version is much less forgiving when it comes to rotation, locking, random generator, etc... It also has a single preview, no hold piece and no hard drop. At high level, a major challenge is to move pieces fast enough to reach the sides, including a technique called "hypertapping" where you mash buttons extremely quickly to move the pieces faster instead of just holding a direction.
Modern Tetris is much faster. At high level, you get to "20G" mode, where the pieces drop immediately and you have to rely on the lock delay and rotation system to move pieces around. Later versions introduced T-Spins as an alternative to Tetrises for high scores and attacks in multiplayer.
I would argue that TGM is closer in spirit to NES Tetris than to modern Tetris. In both NES and TGM, the speed of the game is extrinsic - it comes from the game itself, forcing you to play faster and faster.
In modern Tetris, the speed is intrinsic - it comes from the player skill. Even with forced 20G the rotation system and the wall/floorkicks mechanism makes it much more easier to both move and recover from screwups.
Tetris Party for Wii is well liked in our house. Adds some fun game modes and twists on the original gameplay.
Tetris 2 for SNES is different (goal is to clear certain colored blocks for each level instead of just not dying). My wife and I found it to scale well for our differing video-game skills, as I tend to be better than she is.
Yes, the newer rules (seeing multiple upcoming pieces, swapping pieces, pieces being shuffled instead of completely random, being able to move a piece once it's touched the ground) seem almost like cheating compared to the NES version.
https://tetris.wiki/Tetris_The_Grand_Master