To wait means to not do an action until a future event or condition is met. It can also mean to be in a state of expecting something to happen in the future, but not taking action to make the event come sooner.
Await is a verb that means to wait for something or to be waiting. Also, it can mean to have something in the future be waiting for you, or in other words, something is in store. Await has several obsolete and archaic definitions. It carries with it a general formal feeling or one of seriousness.
Both words can be used with objects; however, if wait is used with an object it requires a preposition (e.g., I wait for the box vs. I await the box). Wait is the only one that can be used without an object. It is also used in many phrases with slightly altered meanings, like to wait up is to stay awake for some event, and if one can’t wait it means he or she is extremely excited.
In short, use await with an object and if the tone required is more formal.
So he waited, then singled out two people who weren’t standing. [USA Today]
The 184 students, who were refused promotion to 12th standard due to poor performance in the previous class, await the decision of their respective schools. [The New Indian Express]
According to PA Foreign Minister Riyad al-Maliki, the authority is still awaiting the court’s authorization in order to submit the documents, which it may do by the middle of June, Israel Radio reported. [The Times of Israel]