Hi guys! 
I've some question about climb speed: can I climb any type of vertical surface, even if there aren't footholds or protuberances?
And over the ceiling? In the positive case, what's the difference between climb speed and spider climb?
In my opinion I can climb only climbable surfaces by an Athletic check, but a friend of mine saysclimb speed allow to climb all vertical surface, like mirror or similar. 
Thanks for asking.
Well, that's nonsense. It's perfectly reasonable to require that a character has a means of employing a given movement mode before they can benefit from it.
What is giving them a climb speed, though? Because a magic spell like Spider Climb can be expected to overcome certain constraints of common sense because MAGIC. Does the source of the speed imply that they can stick to things like Spider-Man, or not?
The only thing a climbing speed guarantees is bypassing certain extra movement costs to climbing. The DM is otherwise free to ask for an ability check for especially difficult climbing surfaces, climbing speed or not. To note, the Spider Climb feature ignores any ability check when moving through those difficult surfaces described above.
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the DM's option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. Similarly, gaining any distance in rough water might require a successful Strength (Athletics) check.