How do I compare two files with a shell script?

diff sets its exit status to indicate if the files are the same or not. The exit status is accessible in the special variable $?. You can expand on Ignacio's answer this way:

diff --brief <(sort file1) <(sort file2) >/dev/null
comp_value=$?

if [ $comp_value -eq 1 ]
then
    echo "do something because they're different"
else
    echo "do something because they're identical"
fi

In bash:

diff --brief <(sort file1) <(sort file2)

Whilst diff is a perfectly fine answer, I'd probably use cmp instead which is specifically for doing a byte by byte comparison of two files.

https://linux.die.net/man/1/cmp

Because of this, it has the added bonus of being able to compare binary files.

if cmp -s "file1" "file2"
then
   echo "The files match"
else
   echo "The files are different"
fi

I'm led to believe it's faster than using diff although I've not personally tested that.