Key difference: Interest relates feeling. Hobby relates action.
According to Dictionary.com, an interest is the feeling of a person whose attention, concern, or curiosity is particularly engaged by something, or something that concerns, involves, draws the attention of, or arouses the curiosity of a person. Basically, an interest is something one likes to do or something that one has a feeling for. For example, she has an interest in poetry, or he is interested in watching the game.
A hobby is something one likes to do in his past time, a spare-time recreational pursuit. It is an action, something one does. Usually, hobbies need devotion; it is something that has to be done systematically. For example, stamp collection or playing an instrument.
Interest has a broad scope which includes hobbies. If one has an interest, they may or may not develop it into a hobby. If someone is somewhat interested in something, he may not pursue it actively. For example, he is interested in sky diving, but he hasn’t gone even once. On the other hand, one will only pursue a hobby if he is interested in it. For example, she is interested in classic literature, her hobby is to collect and read old books, or, he is interested in birds, his hobby is bird watching.