We have been on both ships and here is our opinion.
Strictly going by the ships, I would give the nod to Conquest. The Conquest had her maiden voyage in 2002 so she is a bit newer than the Triumph. She is also a slightly bigger ship. Add to that the fact that the Conquest has a Mongolian Grill (fantastic) and also the best fish and chips at sea in the Sur Mer restaurant...I say you can't go wrong.
For location, New Orleans is worth the drive. We had a great time before the cruise exploring the French Quarter, gambling at Harrah's casino, and enjoying late night beignets at Cafe Du Monde.
A tough decision for you to make. Hopefully some of this information will help. Either way, you are in a for a wonderful vacation. What type room are you going to get? Window or Balcony?