From a strict descriptivist standpoint, your "the car needs washed" construction is not grammatically incorrect - if we're talking about the Central Pennsylvania flavor of English, that is. But your friend is right that this construction is not found in more standard varieties of English, and can be considered incorrect in formal writing.
Anyway... I think the difference between The car needs to be washed and The car needs washing is the degree of cleanliness that will thus be achieved. The -ing version just says the car needs to get some water applied; the end result could be a clean car, or just a slightly cleaner car. The 'to be' version, on the other hand, states that the desired end result is a completely clean car. Which one is the better equivalent for the "needs washed" construction is something you can answer better than I can.
I will note that in most cases, the difference between the -ing and 'to be' versions of a phrase is pretty subtle, and context is king: if your mom says "your room needs cleaning", you'd better assume she wants a sparkling-clean end result, grammar notwithstanding.