There are some pretty big difference for playing games on the two models. Actually, both have pros and cons.
The 2022 model with the A15 chip has a decent performance upgrade over the 2021 with the A12, but the 2022 lacks a fan.
For a lot of games, the 2022 model can sustain higher graphic settings and overall looks better than the 2021.
The problem is with games that really push the GPU limit of the Apple TV, since the 2022 ATV4K3 doesn't have a fan, the 2021 ATV4K2 can actually outperform the ATV4K3.
Some people (including myself) have added heat sinks to the top of the 2022 ATV4K3 to try to keep the temps down when playing games.
The recently added game, Wreckfest, is a good example of where the 2022 ATV4K3 looks amazing, but only for the first few minutes of playing with the high graphic settings, then it starts thermal throttling and the game becomes really unstable, frame rate drops, crashes, etc.
The 2021 ATV4K2's fan allows for almost as good settings as the 2022 model, but keeps the frame rate stays flat and stable.
I think Apple screwed up the ATV4K3, they should have included a fan in the 128GB model.
Oh yeah, that is another big benefit of the 2023 model, the larger storage!