First of all, I have to say that I do have an SG Standard with P90s (2012) with a round profile and also a Les Paul Studio (2012) with a 60s neck. For the ease of playability (upper fret acces, weight) - the SG, as you already know, is awesome. My SG's neck is thicker than my Les Paul Studio's 60's neck profile, and it feels more comfortable for me. I really compared the two necks side by side and it is for sure thicker.
As of the Les Paul, I have the impression that it somehow has a rounder sound and also, like Arlum said, it definitly does have more sustain, because of the construction. The Les Paul will take a bit of getting used to if you want to rock the upper frets, because it is a singlecut model, but I love it. Also, depending on the model, they have different weights. There are, as far as I know, three methods that Gibson uses in order to achieve this weight relief: chambering, traditional weight relief (Classic) and modern weight relief (Studio, Standard). There are some Les Paul models, like the custom shops/VOS and the Les Paul Traditional that have no weight relief.
This being said, I couldn't live without any of those two. They are different and they both have their unique range of sounds that you can get. You should go into a music store and check some Les Pauls and some SGs and see what feels more comfortable to you and also what sound you like more.
PS: I know that this picture has been posted here a couple of times