I am not an expert, but after having called out an engineer for my Hoover tumble drier this week, it sounds like your capacitor for the motor. Can be bought cheaply on the web. Problem is trying to reach the motor depending on your make of dryer. My Hoover model motor is difficult to reach.
Expert I saw this week for my drier, said that if it started when drum turned manually, it was capacitor, and if not, it was the motor. Unfortunately for me it is my motor, which has gone! Earlier my neighbour had a look at it and thought that it may just be a capacitor. You are lucky, if this is all that is wrong -unlike me!
Mrs Evelyn Fordyce (Scotland), September 2008