Award is used as both noun and verb. The award is always associated with something good. The award is always given by someone and is related to some sort of achievement As a noun – Award is a prize for recognition of an achievement, As a verb – Award is giving something for recognition of that achievement. Examples:
He got the student of the year award for his all-round excellence in academia.
He is awarded Scholarship of 4 years for his continued good results.
A reward is also used as both Noun and Verb. A reward can be given both for good as well as a bad action. A reward is not necessarily given by someone, it can be intrinsic. Also, it does not necessarily relate to achievement and mainly relates to action or exchange. As a noun – it is a contingent benefit for an action. As a verb – it is something that is given to someone for doing an action. Examples:
He was being rewarded a gift voucher for winning the dance competition.
She will reward you with a doughnut if you bring her flowers.