In "a rather silly question", the 'rather' is modifying 'silly'; in "rather a silly question", the 'rather' is modifying 'a silly question'. There isn't a discernable difference in meaning. However, there are cases where you would choose one construction over the other: With any determiner other than 'a/an', the 'rather' has to come before the adjective: for example, "You asked some rather silly questions" or "She wore that rather risqué dress". The 'rather' in first position option isn't possible here. However, it's more usual to say "You took rather a long time" than "You took a rather long time", but I don't really know why. The first feels more natural: perhaps because 'a long time' is seen as a fixed concept? I'm not sure, though.

To me, as an American, the word "rather" already feels both old-fashioned and British, but "rather a" feels MORE old-fashioned and MORE British than "a rather." As an American, I'd usually use words like slightly, somewhat, fairly, or kind of. Google Ngrams shows the begfore 1905 in America, and before 1930 in England, "rather a" was more common than "a rather". Now, i both countries, "a rather" is much more common than "rather a". But, as everyone else said, the difference is very small.

There is no significant difference.

Thank you Trang Thuỳ, thank you Tyler!

There is really no difference between these two sentences. The difference in tone is SO minimal, that it would be hard to put it into words.