I like the atmosphere and the aesthetics a lot. I'm not a platformer player, so I had trouble with the controls, especially the wall grab when climbing up was very unforgiving to me. Having to constantly press the direction key when grabbing a wall was tedious to me, there should be at least a short grace period where I float when releasing the direction key (and not instantly fall down) Better yet, once the player grabs the wall, unless there is a new input let him continue to grab, that would be more relaxed and less stiff controls, at least for my taste.
When jumping I feel like the player should float a little on the high point of the jump and not instantly being dragged down. Maybe I'm to slow to learn the right patterns though...
A slow wall slide might do the trick aswell, once the player grabs the wall. Give them a little more time to react I would say (again, non platformer player here) Nice work all in all!