My basic understanding of this is that a backup is intended to be more easy
and quick to restore.  It also tends to be on more expensive media (due to
speed and such).  Archives are kind of your third tier of storage.  If you
have the nth copy of a something that is not likely to be used ever, but
needs to be kept for months/years then an archive pool makes sense.  The
archive pool would be made up of your slowest storage solution.  You have to
look at how much you have to back up, how long you will keep this data, and
how likely it is to be restored.  The reason you even have to ask is that
TSM has so many options.

In my environment I don't have archive pools.  Most of my data (200+ megs)
is backed up to disk.  At night I migrate to 3570 tape.  My 3570 storage is
part of a backup rather than archive pool.  I keep 7 versions of a file, and
a last only for 9999 days (the max).  I move and keep so little data that it
does not make sense for me to have a 3rd storage solution and create archive
pool(s) that utilize it.

So, it really depends on your resources, how much you have to keep, and for
how long.  Either way, as they beat into our head in class, you backup and
restore, and you archive and retrieve.