Both words mean essentially the same thing. They provide the connotation of sitting around in expectation of something, but they are used differently.

"To wait" does not take an object. You could simply say, "I waited." If you want to explain what you're sitting around in expectation of, you could use "for":

I waited for my mom to be done.

"To await" does take an object. You cannot say, "I awaited." You have to follow it with more information; you cannot use "for" with "await."

I awaited my mom.

"To await" sounds a bit more formal than "to wait," so in normal conversation you should probably stick with "I waited for my mom."