You asked, ”specifically in terms of maths, science, computer science, music, debate, personality shape up - which school can give him even the huge advantage over the other”. As a matter of fact, toguptasarika, my son was in almost the exact position as your son when applying for secondary schools many moons ago. The only difference being the private school wasn’t Westminster but perhaps a bigger name one in Berkshire, in addition to getting a QEB offer.
If truth be known, there isn’t an iota of difference in the academics, maths, science, computer science, etc between QEB and Westminster (and the Berkshire school). It’s a question of how these schools get around to achieving the results.
And if you take into account the humanities subjects then there’s absolutely no doubt that the big-named private schools are much better especially in subjects like divinity and the languages, both modern and ancient, e.g. Latin and Greek, etc.
In short, I really wouldn’t worry about “the huge advantage which school can give him” because there isn’t any especially when you’re talking about STEM subjects. For example, my own GP, an excellent physician had attended QEB and pre-clinical at Cambridge before qualifying as a doctor in a London medical school.
In contrast, my DS had attended a private school in Berkshire – a school which usually rank lower than QEB in league tables – but emerging with a full-house 12 A GCSEs (including both modern and ancient languages, and divinity) in addition to a full-house 4 As at A-Level and similarly graduating from the same London medical school (with distinction).
So you see, there isn’t any daylight between these two young men as far as achievements in STEM subjects goes and both achieving the same professional qualification at the end of their university career. The difference is that one got the whole thing for FREE
* and it would cost the other at least ¼ £m under normal circumstances. 
But in reality, there are many huge advantages the private school can give your son over the other. For example, in a boarding school environment, as much as 40% of a pupil’s entire career there is spent outside the classroom doing other things minus the academics. There’s music, sports, drama, dozens of specialist societies, debates, plays, speeches and forums by/with external experts in their particular fields specially invited to share their experiences – the list is almost endless!
That’s what you pay your hard-earned money for. Private schools are not bothered about their ranking position in the league table quite unlike QEB where the regime is one of intense pressure on the academics and nothing much else. If you fall short of their expectations, you’re out even if it means getting rid of you after GCSE so that the school can look pretty sitting at the top of league tables. Whilst the elite private schools educate the complete person and not just the academics.
I remember posting this message on MN just over a year ago. Before attending the prep school and later the senior public school, DS was a shy and reserved child who didn’t speak much. Today, he’s a confident young man with no problems taking care of himself even when in the midst of deep Africa and contrasting early last year when holidaying alone in upbeat Florida but had the good sense of removing himself from the USA (losing quite a lot of money for the pre-paid remaining holidays) barely a week before the USA closed all its borders due to the emergence of Covid-19.
I was mightily impressed seeing and hearing him for the first time giving a public speech at the wedding of a close family member about 3 years ago – the eloquence and stance he adopted – and had a few people coming to congratulate me, the parent of a well-spoken and fine gentleman. *
