I have both and shoot a pair of D800 bodies. I use the 24-120 f/4. I had the older 24-120 and on film or a low MP camera, it was a good workhorse and useful range. The 28-300 as an all-in-one is also a useable lens that a friend uses with success on his D800. For me, it was a good range, but not quite long enough and a little too big for me to hand-old on the long end, thus needing a tripod. It's also a bit too slow on the long range for most hand-holding anyway.
As the MP count increased above 6mp, the 2nd generation 24-120 just didn't hold up. So I went to the 24-70 2.8. Big and heavy and a little short for my style of shooting. When I went to the D800, I tried the 24-120 and it was a good fit for me. it is still my go-to lens and favorite workhorse.
If I go out with one lens, this is the lens followed by my 17-35 if I'm working close to pack a second body. This pair covers 95% of my needs. There's a little distortion with the 24-120 and it's easily cleaned-up in ACR.
What I'd do is rent the pair and give them a try for a weekend and see which of the two fits your style and your needs.
