"To wake up" is a verb and "awake" is normally an adjective. Be aware, "to awaken" is also a verb that means "to wake up," but it is more fancy and less often used. Examples I will wake up at 6:00 tomorrow. The doctor said I would be awake during the surgery, and it scared me! The curse awakened the Kraken from the deep.

"wake" is what happens after a funeral. "Awake" = not asleep, "wide-awake and full of energy".

Awake is an adjective, so it describes a state of being conscious and aware of your surroundings. Wake is a verb and describes the action of waking up - going from being asleep to being awake. It can also mean causing somebody else to arise from their sleep. For example, "I will wake you in the morning". Also, "I awoke one morning" / "I have awoken" - these are both past tense forms of the verb "to awaken". The phrasal verb "to wake up" can also be used, either to describe waking yourself up or waking somebody else up from their sleep. E.g. "I will wake up at 8am" or "I will wake you up at 8am". The main thing to remember is that "awake" will always be used as an adjective, but "wake" is a verb.