I lived in Richmond until 5 years ago when we moved to Twickenham - and obviously have enjoyed living in both places as is where I grew up and haven't moved very far. So if you have any more questions please message me and i'll be happy to help.
Much as I loved Richmond in my 20's - I prefer Twickenham now as suits me better with a young child, but still very easy to pop into richmond if I want to go clothes shopping or out for the evening. Twickenham town centre is less busy, there's a couple of great play beaches, good playgrounds and nice cafes which is what I generally use the town for these days! There's also some lovely clubs for little ones (imagination den and Tempo Tots to name a couple). I'm not sure what age your children are but both Richmond and Twickenham have outstanding primary schools - but be aware they usually have teeny tiny catchments! We don't actually qualify for our two closest primary schools as the catchment has got so small but they just opened a new one near Twickenham Green, and I believe another is being built in East Twickenham soon.
If you live in the right postcode you also qualify for the residents ballot for RFU matches at Twickenham- we got some good tickets for the World Cup from that
In terms of secondary, if you have a daughter then Waldergrave (Twickenham) is often rated one of the best in the country.
In terms of commuting, Richmond does have the tube line - lots of people get the train from Twix to Richmond and change to the tube there - and Twickenham has fast trains into Waterloo, as does Richmond, but at Twickenham you have a better chance of getting a seat!
I could waffle on about them for hours but let me know if there's anything more specific as Richmond/ Twix are my specialist area