I love Pro and Con lists, so here you go:

RDM:
Pros:
Great toolkit for casual play (raise and heal)
interesting playstyle
very easy to do well with (I consider it to be the easiest DPS, this could be a con for you)
great aesthetic
Cons:
Performance in hardcore content is limited due to low DPS output compared to SMN and BLM
very easy class to master (could be a pro)

DRG:
Pros:
Great aesthetic
Very engaging playstyle
Provides very good party utility
Great for hardcore content (currently one of the most sought after DPS in end game raiding content)
Cons:
More difficult to master than RDM (though certainly not the hardest DPS in the game)
Has a TON of buttons to hit, making it somewhat difficult for controller users
Frequent animation locks (due to jumps) results in many deaths while learning content.

TL;DR Red Mage is fun to play, and great for casual players. DRG is more difficult and is better suited for more hardcore players.