The 28-300mm isn't at all leaps and bounds bigger than the 100-400mm. They're both roughly the same exact size, and there's a small weight difference. The 100-400mm is 3.6"x7.4" and weighs 3.1 pounds. The 28-300mm is 3.6"x7.1" and weighs 3.7 pounds. Not much difference there at all.
The 100-400mm has the tiniest edge in IQ, but it's tiny. The 28-300mm gives you a beautiful zoom range, no other lens will give you this much range with the type of IQ you can get from the lens. Both will not be spectacular at the wide end, but there's not many lenses of these types that are.
I've shot the 100-400mm, and I own the 28-300mm, both are great lenses, both will not disappoint you at all. However, you have to decide what you want out of a lens, any lens, before you decide on which lens. For me, the 28-300mm is an absolute killer at the zoo; one lens to do it all, that's for me. Excellent IQ, perfect range for that "take one lens to the zoo" solution. I used to use two lenses at the zoo, my Sig 150-500mm OS and my Sig 18-200mm OS. After buying the 28-300mm I find that I use it most of the time, and the 150-500mm OS when I need the extra reach.
The big kick in the pants with the 28-300mm is the price. If you can find one in good condition used, buy it. If you have to pay full price new, that takes a bit more thought. While it's a fantastic lens, I don't think it's quite worth $2500, no matter how good it is and how much range it has. I paid under $1800 for mine from KEH, but good luck waiting for them to get another one at that price. For $1800 I'd buy it all day long, for $2500 I'd wait.
I spoke to fast, I just checked KEH and they have one for $1739.00.. I'd jump on it if I were you. Mine said it was "EX" condition, and it looked brand new when I got it. The only signs of use were a few slight marks on the tripod foot. Otherwise it was perfect.
The 100-400mm is a great lens and has a more realistic price tag on it, but for a lens of that range I'd look at the Sigma 150-500mm OS. You'll lose 50mm on the low end, but you gain 100mm at the high end, and up to 4-stop stabilization. You get 2 stops at best with the 100-400mm. Plus, the Sig is cheaper. Both the Sig and the 100-400mm have identical IQ, but you get longer reach out of the Sig. To me, when it comes down to the 100-400mm L, I'd prefer the Sig all day long and twice on Sunday.