Well, these are not two exact game types...
Real Racing is "real racing" on a racing track just without any extra bonuses.
Asphalt is something like NFS. The graphics is not that good like RR2 and there are Nitro Boosts, you can eliminate other cars by crashing them. There are lots of speed ups and different game modes. And you can easily reach a 600km/h speed which I suppose is not very realistic.
So, I prefer real racing, because the UI is more user friendly, it's more realistic, better graphics and so on.
/e Also Asphalt crashes quite a lot.