more and less
The 22-250 is indeed more cartridge than the .223, the difference in the cartridge case is no lie, more space (for powder), more velocity. That added velocity over the .223 will make it faster and flatter, and easier to hit small targets further out. I'm going to go out on a limb and say that the 22-250 has 100 yds more effective range on the .223 for varmints. To 300 yds, there is not a lot of difference. Out to 400, with the same zero and same slugs, the 22-250 should give about 6" less drop.
That can yield more hits for the average shooter, as ranging at that distance on small targets is a bit less problematic. But wind and ability will limit one's capacity to effectivly use that flattened trajectory. I've been sniping crows here lately with mine, and 300 is about as far as a crow has any real danger from me. The 200 yd shots that I make often enough, could be made with a .223.
Choice might depend on your shooting too. If you're a high volume (say Prairie dogs, though I've not done it yet), shooter, the .223 will go easier on the bore, provided you use some sense on your rate and volume of fire. With the 22-250, if you don't use good sense, I bet you can cook the throat in a day. If you shoot varmints occassionally, the odd crow/coyote/coyote hunt, the 22-250 should last a long enough while.
The OP mentions he will load for it, and that creates another option. There is nothing to say that the 22-250 can be loaded down to .223 velocity and yield longer barrel life. But you cannot load a .223 up to 22-250 levels to be flatter between 300-400 yds.
Much will depend on the assessement of your pal's rifle. Seems a tad high me. He's a pal, shoot him a counter offer if the rifle's good.