A foreword: I don't like adding any bulk to my phone. Therefore I did not consider any of the bigger ones. Before this I got a Caseology Hybrid TPU/Silicone which was too thick on the back, IMO. Then I got the standard cheapo TPU transparent with an S on the back, which I liked for it was thin, but soon became nasty yellowish.
So I got this Invellop. The feeling is great, the soft spot for the buttons (barely visible below) make very convenient to push them. I suppose it has a hard plastic core and this fake leather outside which matches well the note 3. It is really as thin as the TPU, and even snuggier on the sides. While it does not offer protection on the top and bottom sides, I think this ain't too bad. For once, it's very unlikely that the phone will hit there. If it falls it will hit a corner or both of them at the same time. It may be of course source of damages on rocky floors or stairs, corners of chairs, etc. But hey.
The good thing of this lack of protection is that the home button and the capacitive ones are MUCH more convenient to press. Especially the home button, even with the thin TPU I ended up touching the screen quite often be cose of the rising frame around.
Obviously the screen and the camera lens don't touch when lieing the phone on the back or on the front.
Anyway. It's still a cheap cover so let's see how long it last. Picture follow.
Also has a matte Invellop screen protector (0.95 USD for three on amazon) which are very good and great value. Only I hate matte films... Will change it soon.
Sorry for the poor quality of some photos.