"Await" is a formal word which is mainly used in business and other formal contexts. For example, "We await your confirmation". We rarely use this word in our everyday lives.
Note that it takes a direct object: await + noun.
"Wait" is a neutral word which can be used any context. For example, "We're waiting for an answer". It's a very common everyday word.
Note that it takes an indirect object: wait + for + noun.