You can do this from the command prompt if you want, though maybe there is
some software too?

From the terminal:
~$ ls /path/to/desktop/folder > file_home
~$ ls /path/to/external/hd/folder > file_external
~$ ls diff file_home file_external

the above will create two text files (file_home, file_external) in the first
two steps. the third steps compare them and gives an output as to what is
different between the two. a "<" or ">" will indicate whether it was the
left (file_home) or right (file_external) that had the extra file. You can
play around with two dummie directories on your desktop with text files in
them to get the hang of it.

note: file_home and file_external will be created in the directory you are
currently in. you can leave them for next time, or delete them or whatever
you prefer. They will just be overwritten each time you do the above... oh,
and you can just name them 1 and 2 if you want.