"Awake" is usually only used as an adjective, meaning the opposite of being asleep. The verb form of this word would be either "awake" or "awaken", but they both sound a bit old and aren't used very often.
Ex: I'm going to awaken the dog before I go outside.
I awoke after I fell from the bed.
"Wake" just means to bring someone out of a sleep. Its usually used in the expression "wake up".
Ex: My brother needs to wake up, he's been sleeping all day!
I woke up at 7:30 this morning.
"Wake (up)" and "awaken" mean the same thing, but the expression "wake up" is used almost always over "awaken"!