I wouldn't think that there is much to choose between them. I would suggest that the exit results in terms of school destinations will give you a good indication. Both schools send their children off to pretty academic schools.
I have friends with children at both schools and they are all very happy. Newton seems very international in terms of cohort and outlook. Thomas's very sporty and more competitive. But I know more parents from the Clapham school as opposed to the Battersea School, maybe that is more international. It's certainly more royal
if that floats your boat!
I'm sure that you will be happy whichever you choose.