Having lived both sides of the river, I am certain that things are better North of the River. Sure, there is plenty to do in the south, and there are some lovely areas but now that I have lived North and South, it would take a lot to get me to move South of the River
The main problem is that the transport infrastructure of South London is considerably less developed than the North. Ground conditions have prevented a great deal of London Underground expansion into the south – it costs to much to build the tunnels. That’s why South Londoners have to put up with the less efficient (but slightly more reliable) over ground railway services.
Look on a tube-map. Less than 1/3 of the stations shown are South of the river. South London takes up about the same amount of space as North London, but has far fewer Tube routes. Sure, both North and South London have plenty of buses, but who wants to wait an hour for a Night Bus to get you home on a cold rainy London night?