I have children at FP, and we have relatives at Thomas's Fulham. The schools are similar in their curriculum and both offer relatively the same thing - arts are a bit fancier at Thomas's in the younger years but that's only because the lower and upper school share facilities. FP gets slightly better results than Thomas's. But both schools are good schools with mostly the same types of people going to both - people who live mostly in Fulham, which is an international mix of people.
I think its important to note that the Thomas's 'assessment' is not an academic selection and the school has a sibling policy so the school also has all types of children like FP.
Because the Pre-Prep is separate from the Upper school at Fulham Prep, it means the environment in the Pre-Prep is quite sweet. The head, Mrs Steven, has been there for 17 years and is very experienced in early years education. There are some amazing and creative teachers in the Pre-Prep - they do a lot of interactive and imaginative work to teach the children, eg one teacher had an alien's space ship crash land in her classroom and taught lessons around this. Mr Lunnon in the Upper school is amazing. He's very involved in all aspects of the school and knows the children very well. There are tons of activities going on and the student body are very engaged.
What I have found is Fulham Prep tends often to be understated in some of what it does. Its doing everything you would expect a good school to be doing, but it tends to not talk about it as much (they could talk about it more). So for example, they start prepping the students for 11+ exams very gently early on. They don't make a big fuss about it but seem to ease the children into the next steps.
Also, we have found the children to be overall quite sweet and kind at Fulham Prep. My children have lovely friends. The parents are also quite down to earth, friendly and engaged.