Tetris Axis, the earlier one, is by Hudson and is heavily based on Tetris Party for the Wii and Nintendo DS. I believe it includes all the extra game modes from Tetris Party, some of which are pretty fun. If you have no familiarity with Tetris Party, then maybe Tetris DS is the next most similar thing, in terms of all the reimagined games you can play using Tetris blocks. Axis does have 3D support; the backdrops are cityscapes and things that are 3D, and the blocks themselves are also cubes rather than flat sprites. Honestly, I don't like how the cubed blocks look, but it does add a tiny bit of additional 3D to the mix.
Ubisoft's Tetris Ultimate kind of bores the hell out of me aesthetically, it's just very clinical and ... blue. The game modes are far more limited than Axis, it's mostly just Tetris, but there are a couple of battle modes with garbage blocks and power-ups and things, sort of like Tetris Attack. It has some 3D, basically just a (blue) backdrop behind the play field. The blocks are flat 2D sprites.
It's also worth considering Puyo Puyo Tetris, which was Japan-only on 3DS but has an English translation here on GBATemp. You can play standard Tetris on it if you want, or weird combinations of Puyo Puyo and Tetris. I'm not into Puyo Puyo myself but I still enjoy firing up PPT sometimes just to have a quick round of Tetris. In terms of pure Tetris game modes, it's probably the most limited of the three, because most modes are based around combining it with Puyo Puyo. Aesthetically, I like how this game looks the most out of the three, with its colorful, cartoonish style. It has 3D, with multi-layered backdrops as well as foreground UI elements that pop out.