We were in the exact same position as you 3 years ago.
We looked at both schools on 2 separate occasions (including the open day).
Our son also spent a day with each school and was assessed during the visit.
There was nothing in it as both are excellent schools.
Pilgrims seemed to have a real passion for music demonstrated by the number of scholarships achieved (1 in 3 kids from what I remember) and also their enthusiastic music teacher. Although it is close to Winchester College, the intake was approx the same as Horris Hill. Long list of clubs/activities for kids to get stuck in to.
In the end we chose HH as in our own opinion it was right for us and our son.
Naturally, this might differ for your own child but it was right for our case.
As I said earlier, there was nothing in it as both are excellent schools. We liked the smaller community feel to HH and although an equal number of kids go on to Winchester, the ones that don't chose other schools like Eton & Radley in good numbers. HH is still affiliated with Winchester - see the list of governors.
Pilgrims did say that if for whatever reason we changed our minds then they would welcome us at anytime - that went a long way in my mind but our mind was already set by that time.
I really hope that this helps.
For all those that have nothing else but to distract from your question:
I am only trying to help another parent with their concerns/choices so I will totally ignore all the haters out there - I have heard all the negative comments before therefore, please direct any negativity towards someone that cares.