I am a Hampton parent - two boys there, one is an academic geek, one is not quite as academic as his brother but far more of an all rounder. Neither are A team sport material but both are very happy, both love it. As a parent my overriding feeling is that it's very much a school that prioritises what is right for individuals, encourages excellence but also encourages the boys to be kind and to support each other.
I can't comment on St Pauls in all honesty because we didn't apply, I've never even looked round it, it would have been too much of a faff for our DC to get to and from where we live. That said, I think you're probably right @Rey1979 that St Pauls would be perceived as the more 'prestigious' school to get into and your comment about a 'prized possession' is probably correct.
However, I think Hampton has a very down-to-earth feel, and I suspect this is partly because it's not viewed as a 'prized possession' by many London parents (Westminster/Kings/St Pauls are very much the top 3, not necessarily in that order; Hampton is outside of that top 3).
But as @Hatty999 says please think about your child and best fit for them. It's so easy to get caught up in league tables but at the end of the day it's about where your child will feel most comfortable. If they are happy at school and supported at home they will achieve.