Technically, use is a verb, and usage is a noun, but, and although it's looked down upon by some, speakers today tend to shorten usage to use:
Verb: I use a computer to type my essays.
Noun: What's this word's usage? ~ What's this word's use?
That's not entirely correct. "Use" is not *only* a verb. It's also something akin to a gerund at times. Even in the example you cited, without more context it's not clear as to what you're questioning. If you were to replace the word "use" in the second sentence with "purpose", it's not the same thing anymore. Purpose != Functionality. One word describes the "how to" of something, the other simply questions the thing's existence to begin with.
"I would call him back, but what's the use?" In this case it's actually wrong to use the word "usage" because you aren't questioning the functionality of an item or a process; you're questioning the point or the motivation behind it.