JoJo, I guess areas north north and south of the river have had some gentrification in reality.
Historically house prices were and remain cheaper south of the river. This doesn't apply to all areas and all houses, but it often is the case, that you can get a similar house in a similar area for less, south of the river. In the past, this was even more the case. Whether higher house prices make an area 'better' or actually being a bit more affordable is 'better' is very much a point of view.
Over the years I've known a number of north Londoners move south when starting out, because they could get something as a starter or certainly more for their money south of the river. As others have mentioned, there are big differences when you look east and west too. Some people might happily go from north-west London boroughs to the south-west but would find either the north east or the south east less appealing.
I think with transport the south has been in catch-up. Clearly the tube didn't extend as far south as it went north (right into Bucks and Herts and beyond what became the M25) but southern services have improved with things like ThamesLink and also Oyster has extended beyond the tube into south London too, making the whole transport system feel more unified and perhaps making south London feel more 'part of it'.
There's certain't lying a different feel south of the river. I think north Londoners often notice it. All of Lomdon is ethnically diverse, with some areas having more or less of different groups. And some ethnic groups have been in London in large numbers for longer than others and so perhaps the history of those areas is more permeated by shops, businesses etc which give a different feel.
Schools is an interesting one. Most Lomdon boroughs have non-selective education. I guess the parts of Kent and Essex which fall within London still have selection and some places like Sutton and Kingston and Barent/Enfield have some superselectives. There are plenty of good independnets on both sides of the river and competition is more fierce in Lomdon and the south east for places in sought after schools than elsewhere. Might it be right to say a higher proportion of the really old schools (often public schools with boarding) are based in north London than south? Loads of great, top ranking day schools in both though and further out into Home Counties too,
When you look at the edges of London, so around the M25 and just outside it, Surrey in the south, Bucks and Hertfordshire in the north are probably the most espensice overall. Perhaps historically more people commuted into Lomdon from the northern counties too (although Surrey and parts of Kent big commuter belt too) and the northern Home Counties with their tube lines in the areas just above the M25 have extended that sense of being part of Lomdon, even if not living in a Lomdon borough.