I've known gzip for years, recently I saw bzip being used at work. Are they basically equivalent, or are there significant pros and cons to one of them over the other?
Gzip and bzip2 are functionally equivalent. (There once was a bzip, but it seems to have completely vanished off the face of the world.) Other common compression formats are zip, rar and 7z; these three do both compression and archiving (packing multiple files into one). Here are some typical ratings in terms of speed, availability and typical compression ratio (note that these ratings are somewhat subjective, don't take them as gospel):
decompression speed (fast > slow): gzip, zip > 7z > rar > bzip2 compression speed (fast > slow): gzip, zip > bzip2 > 7z > rar compression ratio (better > worse): 7z > rar, bzip2 > gzip > zip availability (unix): gzip > bzip2 > zip > 7z > rar availability (windows): zip > rar > 7z > gzip, bzip2
As you can see, there isn't a clear winner. If you want to rely on programs that are likely to be installed already, use zip on Windows (or if possible, self-extracting archives, as Windows doesn't ship with any of these) and gzip on unix. If you want maximum compression, use 7z.
Rar also has downside that, as far as I know, there is no free software that creates rar archives or that can unpack all rar archives. The other formats have free implementations and no (serious) patent claims.
The algorithms have different time, memory, space tradeoffs. Bear in mind these algorithms were written quite a while back and your smartphone has many times more CPU than desktops of those days.
Your pick is between universality (.gz) and a bit more compression (.bz2). Only you can say whichyou care about more.
One advantage of .gz is that it can compress a stream, a sequence where you can't look behind. This makes it the official compressor of http streams. I needed to use gzip once because of that, but unlikely you'll need to think about it.
Here is a list of sites that test compression algorithms, to find just bzip and gzip you will have to do some digging, but most sites will list characteristics of the algorithms. This way you can compare what is important to you, size (compression ratio), time, memory, cpu.
Per http://tukaani.org/lzma/benchmarks.html , gzip compresses twice as fast as bzip2, and decompresses ten times as fast.
Eg for use with s3 caching, on travis etc, where you want speed of compress/decompress, not just small sizes, gzip might be a good trade-off.