Very much personal view....
We visited both and are very local to both of them so have lots of friends with DDs at them.
Overall, we found WHS more relaxed and diverse than PHS on a number of fronts. Probably linked (at least partly) with the different locations and where students come from ...more of a Wimbledon / Kingston vibe at WHS, more Putney / Fulham / Chelsea at PHS (not to say that is where most of the girls come from, but it lends itself to a different vibe imho). Extra-curricular / clubs very strong in both - more focus on sports at PHS (though WHS coming up in the ranks). We also preferred the WHS curriculum (including Civil Discourse from Y7 and PPE in Y10).
Both academically strong, though WHS tends to be seen a having a bit of an edge.
Links with KC are a plus for WHS.
Location-wise, WHS in the midst of Wimbledon and close to the village, which the DDs really like as they grow up. PHS in a more residential area / more out of the way.
Overall, we just got the vibe that it would suit our DD better (we also discussed with her prep and they thought the same). It struck as an academic school which encouraged and rewarded curiosity and love of learning and was warm and inclusive.
We ended up not applying to PHS (you have to limit numbers somewhere, and there are great schools in our corner of London). My DD got an offer from WHS and decided to take it, she is now happily there in Y7.
Feel free to PM me if you have any further question.