Copy on Write and auto-compression seem like they will free up a bit of space.

That benefit is more theoretical, than practical. The majority of content you have on your drive will be content that is already heavily compressed at the file format level (e.g. images, movies, music) and where additional compression has no effect. So don’t expect any big savings out of the box, you get about 5-15% saving on a regular install.

The real benefit are in the additional functionality it provides, e.g. you can use copy-on-write to make backups of large games before trying to mod them, backups that you might have previously avoided due to 100GB copies taking so long, with BTRFS it is done in a second.

As for BTRFS snapshots you seem to have the wrong idea what they are for. With them you can create backups of your whole filesystem within a second, so you can have full backup in case something goes wrong. With the snapper daemon you can easily automate that make new snapshots every hour and expire them when no longer needed. That’s a really useful thing to have and with BTRFS it’s basically free.

Another big advantage is the ease of partitioning, or the lack of a need for it, with BTRFS subvolumes all your volumes can share the same free space, while still being addressable as separate file systems, e.g. you can have subvolumes for / and /home, without having to guestimate how much space you might need in the future. Can also be abused to install multiple distributions on the same filesystem.

Also checksums, BTRFS will tell you when your data goes bad.