If you want to count the number of lines that are different use this:
diff -U 0 file1 file2 | grep ^@ | wc -l
Doesn't John's answer double count the different lines?
diff -U 0 file1 file2 | grep -v ^@ | wc -l
That minus 2 for the two file names at the top of the diff listing. Unified format is probably a bit faster than side-by-side format.
diff
If using Linux/Unix, what about comm -1 file1 file2 to print lines in file1 that aren't in file2, comm -1 file1 file2 | wc -l to count them, and similarly for comm -2 ...?
comm -1 file1 file2
comm -1 file1 file2 | wc -l
comm -2 ...