I'm facing the exact same problem! Sometimes W works, sometimes it doesn't. I copied the demo Free Climb Wall to my own scene, and it would not work there as well. The character only moves upward when A or D is pressed with W.
Can an object have two layers or two Object Identifiers? Technically an object should not have 2 layers... but I found an issue with the Free Climb package of the Climbing pack.
Allowing the player to climb allows it to climb on any surface. There is an option to limit them by their layers, like...