Zip makes different tradeoffs. Its compression is basically the same as gz, but you wouldn't know it from the file sizes.
Tar archives everything together, then compresses. The advantage is that there are more patterns available across all the files, so it can be compressed a lot more.
Zip compresses individual files, then archives. The individual files aren't going to be compressed as much because they aren't handling patterns between files. The advantages are that an error early in the file won't propagate to all the other files after it, and you can read a file in the middle without decompressing everything before it.
I'm the weird one in the room. I've been using 7z for the last 10-15 years and now .tar.zst, after finding out that ZStandard achieves higher compression than 7-Zip, even with 7-Zip in "best" mode, LZMA version 1, huge dictionary sizes and whatnot.
If you download and extract the tarball as two separate steps instead of piping curl directly into tar xz (for gzip) / tar xj (for bz2) / tar xJ (for xz), are you even a Linux user?
Bzip2 compression is often surprisingly good with text files, especially log files. It seems to "see" redundancies there - and logs often have a lot of it - far better than gzip and sometimes even lzma.
Anyway, if I saw a bunch of tar.bz2 files, that's what I'd expect to find in them.
I unironically used xz for a long time. It was just eazy and all around very good compress. A close second is 7zip because I used it on windows for years.
I still wonder what that's like. Somebody must still occasionally get a notification that SOMEWHERE somebody paid for their WinRAR license and is like "WOAH WE GOT ANOTHER ONE!"
Compatibility aside, I'd say that .tar.pxz aka .tpxz is probably my vote.
LZMA is probably what I'd want to use. xz and 7zip use that. It's a bit slow to compress, but it has good compression ratios, and it's faster to decompress than bzip2.
pixz permits for parallel LZMA compression/decompression. On present-day processors with a lot of cores, that's desirable.