As the spell description states, GMs can be a bit more liberal with how they interpret the results of Grease. A save would be keeping more in line with the spell, but a climb check wouldn't be out of the question.
The reflex save can be thought of as reacting to the hand/footholds suddenly becoming slippery. They fail, and they lose their grip and plummet as they slide off the wall. Success means they hold on (immobile).
The climb check as a direct result of the spell could be if the target already has a solid grip, which is unaffected by the magic. Movement (which maybe the target was in the middle of) would require the check to see if they can stay on the wall, with at least the -5 penalty for a slippery surface (maybe even -10, considering the wall as a creature who suddenly got the +10 to their Escape Artist).
Either way, further movement would require a climb check at penalty. As per the spell description, going with the reflex save is the more canonical answer.