My DD was at GHS (now at uni). She started fairly shy, middle of the year group roughly and not particularly sporty/musical/into drama (but had other interests and liked sport even though she wasn't 1st team standard).
DD absolutely loved the school and in contrary to what you would expect found it laid back in terms of pressure and relaxed. Things like she said she never remembers anyone ever getting a detention or having to do a re-test. The pastoral care is brilliant, lot of very approachable and caring teachers who really know the girls. Loads of opportunities to try new things and DD even represented her house for the monopoly competition and gingerbread house making!
It is an academic environment, all the girls want to work hard and do well. DD liked to try her best in everything and developed a very strong 'work hard, play hard' attitude which she very much still has now she is at uni. For me, the friendships they make are one of the best things about the school. DD is still very good friends with the GHS girls and I can imagine they will be lifelong friends. They are incredibly supportive of each other which is lovely to see.
Only things DD moaned about were petty uniform rules (but you get that at every girls' school!). I thought the sixth form, while results and uni offers are brilliant, was a bit of a let down in some ways. It was quite academically focused and not all that different from younger years. Things like the school wouldn't put on a leavers ball for them and left the girls to organise it, refusing any involvement. St Cats and RGS both put on wonderful events for the U6 leavers and have a much more celebratory tone, GHS felt a bit like an anticlimax! This is pretty much the only thing though I can think of which we have been disappointed with. It does show how GHS differs though, it is a town day school compared to St Cats which is more traditional, country (boarding) atmosphere.
DD was ready (but sad) to leave and for a change by the end of Upper 6th, I guess that is what you'd expect though after 14 years at one school! Some girls clashed with the atmosphere and ethos of high standards but only about 6/100 left for sixth form which shows these girls were the minority. She looks back at her time there with a lot of pride though and there was always a really special atmosphere in the school and sense of belonging as a 'GHS girl'.
The girls all seem to make friends with boys from RGS in year 10/11+ through music events with RGS and general studies in the sixth form. Lots of parties and socialising in the Sixth Form (and hard work!).
The St Cats new facilities are stunning. The ethos seems quite different- e.g. everyone out playing lacrosse or supporting their house and a bit more religious/traditional than GHS. They have a very strong house system throughout the school. They mix with Eton/Charterhouse more I believe.
Really depends what you want- both are brilliant schools and results wise your DD would probably do the same. Just very different style of school with one being town the other countryside.