The normal rules for climbing state that:
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. At the DM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check.
(noting that this is a stealth correction to the original wording of the rules, which accidentally suggested that your listed climbing speed is irrelevant if your walking speed is faster)
The Thief rogue archetype has the ability Second-Story Work which provides the following benefit:
When you choose this archetype at 3rd level, you gain the ability to climb faster than normal; climbing no longer costs you extra movement.
The Athlete feat, as one of its benefits, offers the same effect:
Climbing doesn't cost you extra movement.
As far as I can tell, a 3rd level Thief or an Athlete functionally has a climbing speed equal to their normal walking speed, but they definitely don't actually have a climbing speed. Unlike in previous editions, having a climbing speed doesn't seem to offer any side benefits, such as a bonus to skill/ability checks made to climb in difficult circumstances.
Is there any mechanical difference between having an actual climbing speed equal to walking speed or having the ability to climb with your walking speed without penalty? Are there any other abilities or effects a character might be subject to where the distinction is meaningful?