Only show what's new in a file
Solution 1:
If you want to get the additions only you can filter the output such as:
$ diff test.log.before test.log.after | grep "^>" | cut -c 3-
However, you will lose the deletions if any.
Solution 2:
If you always do a before and after such that the end of the after file will contain the new content then you could use comm.
$ cat before
aaa
bbb
$ cat after
aaa
bbb
ccc
$ comm -3 after before
ccc