От man diff
:
diff - compare files line by line
ОПИСАНИЕ
diff [OPTION]... FILES
ОПИСАНИЕ
Compare files line by line.
...
-r --recursive
Recursively compare any subdirectories found.
Итак, diff -r dir1 dir2 > diffs.txt
или просто diff dir1 dir2 > diffs.txt
, если вам не нужно, чтобы он был рекурсивным.