Also, to me, TGU is not a push. Someone here said not so long ago it is very similar to a "vertical carry", which I think is a good definition of the movement. It helps the press for sure, but I still stand to my (probably wrong) idea that it will not improve it alone, if you don't actually press.
To me, TGU gives you resilient shoulders, the press gives you strong ones and both develop an iron solid mid-section, if performed with a sufficient weight (expecially the get up for the sheer amount of time under tension and the continuous shifting of the body under the load).
I find what
@natewhite39 says about awareness of a weight over your head to be very true. Every loaded movement is dangerous to some extent, and I feel like the Turkish get up si the "worst of them all": it is a constant joint rotation, performed under load and while the body changes six stances (lying, elbow roll, seated, three points stance, kneeling, standing) and back again for a total of eleven very articulated movements. It is the main reason why I approach the get up with a lot of caution... This, and because you really can't predict where the bell is going to fall, if it happens. The press is much more forgiving in this sense.
If done regularly and correctly, though, it forces the body to adapt to a variable stress like no other exercise, other carries excluded (which, now that I come to think about it, could be why Dan Jon stresses the importance of loaded carries).