To me, as a US English speaker, there is NOT a clear quantitative difference between "quite a few" and "many," but "quite a few" can have a more euphemistic quality in some cases.

For example, if John has had 20 shots of vodka, saying "he drank quite a few shots" might come across as slightly more polite than saying "he had many shots" or "he had a lot of vodka". (But 20 is 20!)

"Quite a few rapes are committed by African-American men" may sound less politically incorrect than "Many rapes..." (But the statistical percentage, whatever it may be, is still the same.)

And "As an actor, I've received quite a few awards" seems less boastful than "...many awards." (But the total number of awards is the same.)

So, in short, I agree with Hanna that "quite a few" is a synonym for "many", but with a spoonful of understatement -- and IMHO, it's the same in US and UK English.