When do you need to make an Strength (Athletics) check to climb when you have a climb speed?

Question

The uncommon Potion of Climbing has this effect:

When you drink this potion, you gain a climbing speed equal to your walking speed for 1 hour. During this time, you have advantage on Strength (Athletics) checks you make to climb.

Looking at what a climbing speed does, the Monster Manual (p. 8) states:

A monster that has a climbing speed can use all or part of its movement to move on vertical surfaces. The monster doesn't need to spend extra movement to climb.

I always understood this to mean that the monster can just "walk on walls" without needing any climb check, and so would you, had you imbibed a potion of climbing. So, I have two questions:

1. Do you need to make an Strength (Athletics) check when moving with your climb speed?
2. If not, under what circumstances must you make such a check if you have a climb speed?

Answer 1

When the outcome of an action linked to climbing is uncertain

The rules for ability checks, in general, state (emphasis mine) :

The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.

This means that whenever the result of an action linked to climbing is uncertain, you should roll a Strength (Athletics) check, which would get advantage from the potion's effect.

Looking at the rules for climbing speed you quoted, the climbing speed itself doesn't nullify the challenge of climbing on something, it simply allows you to climb faster and further than you normally would with just your normal movement speed.

That's why there's also the second effect of the potion : while, as written, it doesn't make you spiderman, it still makes it easier for you to climb on stuff. Up to you to decide on what flavor you want to give to the advantage (does it give you the skills of a veteran climber? does it make your hands somewhat stick to walls? does it let you see where it's easier to grab the surface?)

Moreover, the rules quoted from the Monster Manual apply to monsters with climbing speed, which does not implicitely make it apply to player characters. For those, we can only use the general rules for how climbing works.

It is also good to note that the effects from the potion are drastically different from spell effects such as Spider Climb : the spell allows to climb on surfaces in many situations where you normally wouldn't be able to, while the potion only increases the odds that you'll manage to climb on a surface you would normally be able to climb.

Deciding what counts as a "check you make to climb" is up to the DM. The typical roll for this would be for climbing difficult walls, trees or other elements that would make for a difficult climb for one reason or another.

Other typical rolls linked to climbing would be to resist some external force trying to make you fall from the wall you're climbing or currently holding on to, or trying to grab someone (or something) while keeping your grip. Do note that those are in a grey area, as there is no clear definition of what is and isn't considered in "climbing". As a player, best practice is to discuss any of those specific situations with your DM beforehand.

Answer 2

The important part of the climbing speed description is, "...doesn't need to spend extra movement to climb."

Normally climbing costs extra movement. The potion in question eliminates that extra cost and grants advantage on climb checks, but otherwise doesn't alter the need to make climb checks.

Answer 3

Under what circumstances do you need to make a check?

Whenever the DM calls for it

As Matthieu says, whenever the success of a climb is uncertain, the DM asks for a roll. This could be because of the surface itself (sheer, slippery...), the environment (windy, rainy, foggy...), or the climber (heavily encumbered, only one hand free...).

When the surface is more inclined than vertical

Even on the most-climbable of surfaces and under the most favorable of conditions, RAW, a roll would be called for every time you were moving on a non-vertical surface.

As the OP cites, the MM says (emphasis mine):

A monster that has a climbing speed can use all or part of its movement to move on vertical surfaces.

However, many monsters in the MM have the spider climb ability, which says (emphasis mine):

The [monster] can climb difficult surfaces, including upside down on ceilings, without needing to make an ability check.

Thus even beyond a DM's ability to describe the environment and call for a roll, any time the surface being climbed is more difficult than vertical - that is, an overhang, or horizontal but with the creature underneath - a roll is called for. The potion of climbing will give you advantage on the roll but will not obviate the need for it - whereas slippers of spider climbing will.