What is the difference between Desire and Will?

I have the following exercise in my schoolbook:

As you can see, I had to match the phrases with the respective gaps.

My question is: Why can I not swap will and desire?

Here is my research (I used the Cambridge Dictionary):

Will - 1) what someone wants to happen 2) the mental power used to control and direct your thoughts and actions, or a determination to do something, despite any difficulties or opposition

Desire - a strong feeling that you want something

Having done this research, I don't see the difference between these two words and in which context each would be more appropriate.

Solution 1:

There are many definitions for each, but generally the following I think is a half-decent explanation, even if a little simplistic.

Desire is something you want.

Will is something you have.

Desire means wanting a flashy car, or a juicy steak.

Will is the strength and condition you have to start something and follow it through, to take action. It's fitting that another word "willpower" has "will" in it, because in a way it's the strength in you to carry out your actions.

Will is your resolve. resolve
15. something determined or decided; resolution: he had made a resolve to work all day.
16. firmness of purpose; determination: nothing can break his resolve.
Collins Dictionary

So in your example sentences, (I know I'm simplifying) imagine:

"I always feel a desire/willpower to run away."
"I guess I just don't have the desire/willpower to win."

willpower
The strength of will to carry out one's decisions, wishes, or plans.
Willpower

HotLicks' comment under your question probably sums it up best.