I would almost totally agree with everything Musken says. I've stayed once at Pulo and maybe a half dozen times at Puri. Just a few comments.
1. The overall design and ambiance of Puri especially from the main pool area is one of the most beautiful and tranquil man made settings anywhere on the planet.
2. The natural setting is more beautiful at Pulo.
3. As Musken says food is a little better at Puri. If you like Thai food, Puri has some incredible dishes.
4. Spa - I think the therapists at Puri are better.
5. I like the villas better at Puri for the most part, but the Garden villas there lack privacy and feel a little crowded. I've opted for Trisara when the only thing available was a Garden Villa at Puri. The pool villas are Trisara are really, really nice.
6. Ditto on the beaches. Note that the ocean at Puri is a little rough in the off season.
7. Service - ditto except Sandra (AGM?) at Puri is very good and very attentive.
They are both really incredible places. I would be hard pressed to pick just one so I'm thinking of going back to both in April and May.