The schools are both strong but so different. (We looked at both too.) Hampton certainly has better on-site facilities for sport. Both are really good schools and I know lovely boys from both schools who are happy there. All the Hampton parents I know are delighted with it.
It depends what your DS is like.
Both my DSs ended up at RGS - it was their first choice school and ours too. But that's such a personal thing. RGS is quite eccentric and that suited our DC. I couldn't fault the school. They are so happy there. The pastoral care is second to none. It definitely suits a geeky nerdy boy who loves science or music. Hampton may suit a sporty boy more. (Though the super sporty types I know at RGS also seem happy there.) As Lifeis says, RGS stands out among the local independent schools for being particularly unsnobbish about material possessions, status and wealth. No one seems to care. We liked that about it.
For outdoor activity, they seem to spend a lot of time mucking around on the all terrain area, which is quite big. And they have huge, beautiful playing fields at Bradstone Brook. Alsothe cross country team have the advantage of the Surrey Downs on the school's doorstep so they train in open countryside which they enjoy.
One thing to bear in mind is their extra curricular life. RGS might have the edge there. Boys at both schools live very far away from each other. For RGS boys, Guildford soon becomes a meet up spot they can all reach during the holidays. They go bowling, to the cinema, out for dinner, to the Lido or Airhop. They stay behind after school on Fridays to go for coffees or icecreams, browse the book and music shops, catch a film or grab a pizza. It's very sociable. I get the impression from Hampton boys that its location means they are more likely to just go home on the bus and that's it. Public transport to RGS is easier which makes it a bit more flexible when they are doing after school activities. MY DC often stay for after school clubs and can make their way home. With Hampton, I think that if you are at an orchestra practise or match that ends after the school bus has left it's a bit of schlep on different buses. (Depending on where you live.) Just things to bear in mind as they get older.