I would say they have their similarities. Both don't quite fall into the outright sim or arcade niche. Both have a large number of real-life cars and tracks. Perhaps the handling on Grid is slightly more arcade like but I don't think there's that much in it. Both appear to offer a career option of going up through the ranks getting better cars etc.
How are they not remotely similar?
Grid is fun to play and has crash damage it also has a great sense of speed.
Gran turismo has no crash damage and is slow and boring.