They are not "both willpower". They're as different from one another as they are from a DEX or STR save.
I know that's blunt, but it's an important lead-in to any answer to this question. The mistaken premise here is that you are defining these explicitly distinct qualities in terms of an analogy (that each equals willpower in some sense), and then are building a syllogism based on those simultaneous analogies which would suggest that CHA and WIS are the same thing.
They aren't. There is not a stat called Willpower used for all of these purposes. There are two distinct stats used for different purposes. Any investigation which begins with "they're the same, so how are they different?" is going to be challenging.
At their cores, CHA and WIS represent fundamentally different ideas. Charisma is not about force of will but about force of personality. A charismatic character can inspire people to follow them or to agree with them or to believe them.
A sufficiently specified character can produce similar effects on reality itself-- that's how a sorcerer is different from a wizard. The wizard studies reality and the Weave, learns how they interact, and then follows known steps to manipulate the Weave to produce effects in the world. A sorcerer doesn't know any of that stuff, and instead sort of intuitively persuades the Weave to impose an effect on reality in a manner they choose. Sorcerers are filled with magical power and they rely on their charisma to direct it. Wizards are filled with proven knowledge and then apply it.
Wisdom is the ability to discern relationships between things and gather insight into them. Wisdom is about understanding the world around oneself. Wisdom isn't at all about imposing one's will on reality (even for wise characters that want to do so, the wisdom itself doesn't directly cause or direct changes). It's about understanding what reality is and ought to be, along with understanding the implications of things being otherwise.
Spells with CHA saves tend to be about imposing a change on reality, with the target accepting or resisting that change. Think of Banishment: the caster is attempting to literally change the plane of existence on which its target... exists. The target can assert their will to resist that change. It's not about understanding what's happening to them, it's about refusing to let that thing happen.
WIS saves are more about understanding what's going on. When an Illusionist whips up an illusory boulder in the air above a target, the target's wisdom allows them to connect the dots: this isn't how the world behaves, with boulders just appearing in midair, so that boulder may not be real. A high WIS saving throw means their understanding was deep enough to pierce the illusion, while a lower saving throw indicates they aren't totally convinced but not so strongly as to break the illusion.
As an alternative for the boulder case, an INT response might be something more like sitting down and calculating all the specific things about the boulder and its appearance that don't seem to fit with normal reality and then precisely deducing its likely illusory quality. That may be hard to apply in a saving throw situation, though.
There was an amazing analogy posted in an answer to another question on this stack which I was unable to find and reference properly. This partial analogy is not mine, and I would be happy to attribute it to its actual author.
Wisdom is about knowing that tomatoes don't belong in a fruit salad (whether they are technically a fruit or not). Charisma is about making a fruit salad that everyone loved, whether the tomatoes in it belonged or not.