You're right Davidkira. The first 'fast' is an adverb and the second one is an adjective.
You're also right when you say that the two sentences 'are quite similar'. They both have a structure of: Subject+verb+fast. Now, the confusion comes from the verb look as it is used in this example. Look here is not a synonym of see.
That car looks fast= when people look at that car they think that it is fast= That car seems fast.
So, fast in the first example is an adverb because it describes the action. It is his way of driving that is fast, not him.
fast in the second example descibes the car, not the verb. That's why it is not an adverb. And we call it an adjective.
Hope that helps.