Action 2 can shoot 4K (4:3): 4096×3072 so it looks like it captures full sensor image without cropping it to 16:9 ratio. That seems to be handy for post processing if e.g. you would like to add dynamic zoom or the like to standard 4K movie.

Is there anything else where I can benefit from 4:3 mode?

As I understand it, the point is that you can reframe the sensor output to your liking - crop it to 16:9 but not necessarily in the middle.  It seems to be the new way of doing things.  Perhaps it's particularly useful with a camera with a tiny screen where really accurate framing may be difficult on location.

If you want to change the framing in post.

Thank you guys this is in line with my impression too. I have to test this mode and see if it is usable. The only concern is that due to fish-eye lens the distortions on the sensor edges can be uncorrectable. To be honest I would prefer less FOV and less distortions. Dewarp is probably doing the best but still the footage causes some dizziness when camera pans.