Quite is an adverb of degree or relativity. In British
English, it can have two meanings depending on the context:
- (i) ‘a little, moderately but not very’, or
- (ii) ‘completely, very, totally’.
In American English, „quite“ tends to be used mostly
(but not exclusively) in sense (ii), always depending on the context.
Rather has the same sense as „quite“ in meaning (i) above and, as you
note, is arguably slightly „higher style“ or slightly „more formal“ than
„quite“.
In your two example sentences („… quite interesting“ and „… rather
interesting“) there's no difference in meaning out of context. If the speaker
feels there's a risk of his intended meaning of „quite“ being
misunderstood, he might choose to use a different word, e.g.
- it was quite interesting → it was really
interesting
- it was quite interesting → it was fairly
interesting
(Quite) often, „quite“ has no significant intrinsic meaning at all and
just acts as a „filler“ word, which tends to make the style more
conversational.