Just some quick impressions, as someone who played the game on vita and loved it (enough to get the platinum even): If you are one of those people who has lamented the death of the 3D platformer, you owe it to yourself to buy this. It is by far one of the most imaginative and creative games released this generation on any platform. The aesthetic is beautiful and permeates everything, it os simultaneously the visual style and the rules and concept that inform the world and the gameplay. Everything is paper, everything looks and behaves as paper would, and this core concept is what ties all of the mechanics together. You fold down strips of paper with gusts of wind to create platforms, you rip paper open to access new areas, etc. Due to this unified aesthetic the world is very convincing and you really feel transported into a place separate from our own, in the best way.
It also marries the charming aesthetic and presentation with quality core gameplay and platforming challenges. It fluctuates between being a course-focused platformer and exploratory platformer in the best way, with each location usually focusing on one aspect or the other. You'll spend an equal amount of time walking around and enjoying the environment as you will performing tight, challenging platforming sections, and they are equally as engaging. While occasionally a puzzle or section with be obtuse or fall flat, the overall experience is high quality.
As for the new content, there are a variety of sections from the original game that have been retooled for the new controls, as well as some sections that are completely original that usually showcase tools not present in the original release. As a whole, every area is bigger, more open, and contains more secrets and collectibles. However, the game still maintains its relative linearity, and never feels overwhelmingly open. It's just enough freedom so that you don't feel constrained, while always gently pushing you towards the next location or event. There are some pacing issues that come from inputting new areas where there were none in the original release, but I feel that these are only obvious to people who have played the original. If this is your first time with the game, I doubt you will notice. I do think the game loses something by not having the touch controls (something I never thought I would say about a game), but if you never experienced the way touch was implemented the first time around, you won't miss it.
Ultimately, I urge anyone who has an interest in platformers, games with unique aesthetics, or charming and colorful games to give this one a shot. You will not regret it, and it is most definitely worth the $40 price of admission.