@deroberts answer is great, though I want to share some other information that I have found.
gzip -l -v
gzip-compressed files contain already a hash (not secure though, see this SO post):
$ echo something > foo
$ gzip foo
$ gzip -v -l foo.gz
method crc date time compressed uncompressed ratio uncompressed_name
defla 18b1f736 Feb 8 22:34 34 10 -20.0% foo
One can combine the CRC and uncompressed size to get a quick fingerprint:
gzip -v -l foo.gz | awk '{print $2, $7}'
cmp
For checking whether two bytes are equal or not, use cmp file1 file2. Now, a gzipped file has some header with the data and footer (CRC plus original size) appended. The description of the gzip format shows that the header contains the time when the file was compressed and that the file name is a nul-terminated string that is appended after the 10-byte header.
So, assuming that the file name is constant and the same command (gzip "$name") is used, one can check whether two files are different by using cmp and skipping the first bytes including the time:
cmp -i 8 file1 file2
Note: the assumption that the same compression options is important, otherwise the command will always report the file as different. This happens because the compression options are stored in the header and may affect the compressed data. cmp just looks at raw bytes and do not interpret it as gzip.
If you have filenames of the same length, then you could try to calculate the bytes to be skipped after reading the filename. When the filenames are of different size, you could run cmp after skipping bytes, like cmp <(cut -b9- file1) <(cut -b10- file2).
zcmp
This is definitely the best way to go, it first compresses data and starts comparing the bytes with cmp (really, this is what is done in the zcmp (zdiff) shellscript).
One note, do not be afraid of the following note in the manual page:
When both files must be uncompressed before comparison, the second is uncompressed to /tmp. In all other cases, zdiff and zcmp use only a pipe.
When you have a sufficiently new Bash, compression will not use a temporary file, just a pipe. Or, as the zdiff source says:
# Reject Solaris 8's buggy /bin/bash 2.03.