First, define what you mean by "60-day old directory." Do you mean the directory was created 60 days ago or the oldest file contained within it is 60 days old?

I believe that most O/S mark a directory as modified (change it's mod time) only if a file has been added or deleted (this includes moved), but not if a file has been changed, i.e., touching an existing file won't necessarily change the mod time on the including directory¹.

The actual function you would is either rmdir or rmtree from the File::Path module. Unlink won't remove directories, unless "you are superuser and the -U flag is supplied to Perl." It also warns against using unlink to remove directories in all circumstances.

¹ …of course, these same comments apply to directories within the top level directory, and their sub-directories, and so forth. Recursion!