Is there a functional difference between having a listed climb speed and being able to climb at full walking speed?

Question

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?

Answer 1

With certain items, the distinction is meaningful

By default, there is no meaningful distinction between having a climbing speed and being able to climb without spending extra movement, assuming the resulting speeds are equal. However, certain items affect your walking speed without changing your climbing speed:

• Boots of Speed double your walking speed specifically. A Tabaxi PC's climbing speed is unaffected by them, but they could keep climbing if they have walking movement left.
• Horseshoes of Speed increase a horse-like creature's walking speed by 30 feet. (As an aside, the UA Centaur has the Equine Build feature, which makes certain climbs cost 4 extra feet instead of 1 extra foot. RAW, Second Story Work and Athlete would (humorously) negate this.)
• Boots of Striding and Springing set your walking speed to 30 feet.

With these items, it is generally more beneficial to be able to climb using your walking speed without penalty, than it is to actually have a climbing speed. For example, using their movement:

• A Tabaxi PC (Speed 30', Climb 20') climbs 25' (20' + 5', with the additional 5' coming from half their remaining walking speed).
• A Tabaxi PC with Athlete (Speed 30', Climb 20') climbs 30'.
• A Tabaxi PC with Boots of Speed (Speed 60', Climb 20') climbs 40' (20' + 20').
• A Tabaxi PC with Athlete and Boots of Speed (Speed 60', Climb 20') climbs 60'.

The original PHB seemed to imply that simply having a climbing speed was enough to avoid spending extra movement while climbing, even if you weren't using your climbing speed. A stealth change was made to the Climbing, Swimming, and Crawling section around the time that the November 2018 errata came out that clarifies things (D&D Basic Rules 2018, p. 67; emphasis mine):

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.