diff while ignoring patterns within a line, but not the entire line
It isn't exactly what you are looking for since I'm not sure how to retain the dates, but this does solve a couple of your issues:
diff -u --label=file1 <(sed 's/^\[....-..-..\]//' file1) --label=file2 <(sed 's/^\[....-..-..\]//' file2)
Output:
--- file1+++ file2@@ -1,2 +1,2 @@- Some random text foo+ Some random text bar More output here