Both are solid choices. As others have already said, pick the one you like the most.

Currently in the end game meta though, it's worth noticing that DRG is in a more desirable spot since they provide an overall higher DPS contribution for the group.
MNK brings some utility in the form of a stronger (traited) Mantra with a slightly higher single target dps, however it's still lower than what a DRG can provide to the raid in the form of Battle Litany and Disembowel.