"Await" is more formal and/or old-fashioned. Also, they require different grammar. 1) I will wait for you. I will wait for your response. 2) I will await your arrival. I await your response. Basically, "wait for" = "await".

Salsero from Canada has already answered. I'll just add that it's useful in sentences/situations such as... "A surprise awaits you!" (Maybe it's your birthday and someone has bought you a present!). You cannot substitute 'waits for you' in this case.

(I copied this from http://www.englishpractice.com/vocabulary/difference-wait-await/) Difference between wait and await: There are mainly two kinds of differences between ‘wait’ and ‘await’. The verb await must have an expressed object. It is a transitive verb. To await is to wait for something that you expect to happen. I am awaiting her reply. They are awaiting the birth of their baby. She is awaiting a call from her boyfriend. Note that the object of await is usually an inanimate object. For example, we don’t usually await a person, but we can await their call or letter. I am awaiting her response. (BUT NOT I am awaiting her.) The verb wait can be used in different structures. To wait is to stay in one place because you expect that something happen. Wait can be used without an object. We have been waiting for ages. I have been waiting for a bus for two hours. Wait can be followed by an infinitive. The passengers were waiting to board the bus when the bomb exploded. When we use wait, we usually also mention the length of the time we have been waiting. It is not necessary, but it is very common. She has been waiting for a call from her son since yesterday. Before an object, we usually use wait for. The object of wait can be a person or an inanimate object. We are all waiting for you. (NOT We are all awaiting you.) (NOT We are waiting you.)