diff(1) is not the answer, but comm(1) is.
NAME
comm - compare two sorted files line by line
SYNOPSIS
comm [OPTION]... FILE1 FILE2
...
-1 suppress lines unique to FILE1
-2 suppress lines unique to FILE2
-3 suppress lines that appear in both files
So
comm -2 -3 file1 file2 > file3
The input files must be sorted. If they are not, sort them first. This can be done with a temporary file, or…
comm -2 -3 <(sort file1) <(sort file2) > file3
provided that your shell supports process substitution (bash does).