I live in bang in between Leamington and K'worth! Both are nice, but very much depends what you're looking for.
Leamington is a good size town, with decent shopping (lots of big names plus plenty of independents); loads of nice coffee shops, endless places to eat out and lots of bars/pubs. Train station with frequent trains to London/B'ham. Cinema, theatre, art galleries, big library and every type of 'club' going that you might wish for.
There are nice parts to live in and slightly less nice parts - but more because they're studenty rather than because they're grim.
Kenilworth is much smaller - has a nice Waitrose in the centre, a small Boots, but no other big name shops. No clothes shops, lots of charity shops and slightly random shops. The 'new' high street is a bit soul-less, whilst the old high st is lovely but with very few shops on it. There are nice places to eat, but not that many - ditto places to drink. 2 tiny theatres (am dram rather than professional theatre), no cinema or anything else cultural.
In terms of schools, there are good primaries in both (don't know which ones they are though), and Leamington has options for a number of secondaries - some are v good, 1 or 2 less so. What K'worth does have is an excellent secondary.
You don't mention Warwick - have you considered living there? It's got more going on than Kenilworth (in terms of shopping - loads of great indie shops - and eating/drinking), but it's smaller than Leam. But Leam/Warwick are right next door to each other - people from Warwick often talk about going 'into town' meaning Leamington. Great schools there too.
Traffic can be a pain in the backside though - are you moving for work? If so, where are you working, because for me that would have a big impact on where I chose to live.