Remington 'Thunderbolt' brand ammo:
Low-end .22 Long Rifle ammunition, listed at 40 grains, 1255 fps at the muzzle, 140 ft/lbs. at the muzzle.
.22 Hornet ammunition:
Not a particular brand, but a whole new cartridge. Centerfire rifle (I believe pistols have been made for this cartridge at some point, I don't think they are today. I'm sure someone will correct me if I'm wrong). Remington's two offerings are listed as 45 grains, 2690 fps at the muzzle, 723 ft/lbs. at the muzzle.
Two entirely different things altogether. All they really share is the .22 part of the name.
The information I give is straight from the Remington webpage (ballistics section).... so the velocities and energy figures may be a bit 'generous', if you will.
I don't think I could recommend either for deer, if you're halfway interested in a humane kill....YMMV.
-tubeshooter