bash
на помощь.
diff <(sed 's/\(data-refresh-id\)=".*"/\1/;s/\(serverTime\): [0-9]*/\1/' old.txt) \
<(sed 's/\(data-refresh-id\)=".*"/\1/;s/\(serverTime\): [0-9]*/\1/' new.txt)
Итак, в вашем случае
DIFF_OUTPUT="$(diff <(sed 's/\(data-refresh-id\)=".*"/\1/;s/\(serverTime\): [0-9]*/\1/' old.txt) <(sed 's/\(data-refresh-id\)=".*"/\1/;s/\(serverTime\): [0-9]*/\1/' new.txt))