Does rebuilding a clustered index on an IDENTITY column do anything useful?
Since you don't delete rows, no, rebuilding the index will provide no real value outside of updating the statistics for said index, as they are implicitly updated with a rebuild operation.  A more efficient approach is to just run an UPDATE STATISTICS statement instead.

Per Benjamin Nevarez: 


  Rebuilding an index, for example by using ALTER INDEX … REBUILD, will also update index statistics with the equivalent of using WITH FULLSCAN.


To answer your other questions:


  would the table be any different?


Other than located at a different page address, no, structurally it would likely be no different.


  Obviously this is not true if the table has non-clustered indexes as well.


Rebuilding the clustered index doesn't automatically rebuild any non-clustered indexes associated with the table.  To do this, you would need to specify the keyword ALL instead of the clustered index name.  Per MS:


  ...using ALTER INDEX…REBUILD to rebuild a clustered index does not rebuild its nonclustered indexes by default... You have to specify ALL for the index name in order to rebuild all indexes.


And the last question:


  if this table did have additional indexes it still doesn't NEED to rebuild the primary index first if like our table it was all just adds, right?


Correct.  If you never modify a record after it's added to the end of the table, you'd really only ever need to update statistics in order to avoid the Ascending Key Problem as identified by Scott Hodgin's comment below.