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.