Use 'quite' to mean considerably, a lot. "The test was quite hard." "The play was quite long." 'Rather', on the other hand, can mean either 'somewhat' or 'considerably', depending on the context. Usually, it means less than considerably and more like 'somewhat'. "He is somewhat tall, but not incredibly much so." "I was rather surprised when Ted proposed to Anne."

There is quite a lot of noise coming from next door. I would rather go to the movies instead.

Of course both rather and quite have other meanings, but in this sense of "more than a little but not very" there isn't much difference. Rather sounds stronger, "he speaks English quite well" normally sounds less positive than "he speaks English rather well". Besides, rather is typically used when the connotation is negative: "rather bad", "rather boring"...