Does moving while under the effects of the Spider Climb spell actually use your climbing speed?

Question

The spider climb spell states, in full:

Until the spell ends, one willing creature you touch gains the ability to move up, down, and across vertical surfaces and upside down along ceilings, while leaving its hands free. The target also gains a climbing speed equal to its walking speed.

The wording is similar, but not identical, to the Monk's Unarmored Movement Improvement feature at 9th level:

At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during your move.

Monk's, unlike those under the effects of spider climb, do not have a climbing speed. Furthermore, the rules on Climbing state:

Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb or a swimming speed and use it to swim. [...]

Does moving along walls while under the effects of spider climb actually use your climbing speed? If not, does the movement cost extra feet?

Answer 1

It does use your climbing speed if you want it to, and you get to ignore the extra movement cost. If you don't use the climbing speed, you don't get to ignore the extra movement cost

You have the option, evidenced by this line on climbing:

Each foot of movement costs 1 extra foot (2 extra feet in difficult terrain) when you’re climbing, swimming, or crawling. You ignore this extra cost if you have a climbing speed and use it to climb, or a swimming speed and use it to swim. (PHB, Chapter 8: Adventuring, Movement, Special Types of Movement, Climbing, Swimming, and Crawling)

So it is available to you at your discretion because the spell grants a climbing speed, but you don't have to use it if for some reason you don't want to.

If you don't use the climbing speed, then you are subject to the italicized clause in the quote above: because you aren't using the climbing speed to climb, you may not ignore the extra movement cost when climbing.