Hi Tom
We live on HOP and have an 10 month old. We've lived here nearly 6 years now, previously having lived in West London.
• Bringing up kids.
So far, so good. Lots and lots of groups babies \ toddlers, lots of friendly people (out little one's typical week is playgroup on Monday, swimming in Beckenham on Tuesday, Wednesday ad-hoc often the aquarium at the Horniman museum in FH (£6 for a yearly adult pass), Thursday music class at the Honok Oak Pub (they have rabbits there in the pub garden for children), Friday library nursery rhymes.
He'll also head up daily to either Blythe Hill Fields or One Tree Hill (great views over London - Canary Warf etc) with nice playgrounds. Further afield Hilly Fields has the Ravenbourne River running through it (about a 20 minute walk from HOP) or Dulwich Park.
FH has it's outside Gardens and also a big sandpit and green area opposite. And Crystal Palace park is only 3 stops away.
As others have mentioned excellent primary schools.
• Cafés / bars
HOP has two excellent restaurants, Babur (a brilliant Indian - not your typical curry house) and La Querce ( Italian, great food and they love kids in there). HopScotch by the station and Honor Oak pub also heavily geared towards kids during the day, the latter good for an evening drink as is Try on the main street.
• Crime
My only real comparison is Shepherd's Bush, and whilst I have no statistical evidence we feel much much safer here. Bad things can always happen of course, but we find the general atmosphere of the area not in the least threatening.
• Gyms
FH will definitely win on this one. One Gym already, and Forest Hill pools being redeveloped. Of course you can always use the 15 minute walk from HOP to FH as your warm up.
Sure you'll be happier with either, for us FH was a bit busier, HOP has the smaller high street and to coin the estate agent's current favourite term, more 'villagey' (I've no idea how to spell that!).
Good luck!