Hunter8mm: I started using the 22-250 Remington back in 1967, that's nearly half a century ago now.
I have owned and admired dozens of Rifles in this caliber since then.
I still own and use 12 Varminters in this caliber today.
BUT... a while back along came the sensational 204 Ruger cartridge and in factory Rifles I give the clear edge to the 204 Ruger in inherent accuracy!
And that's saying something.
I just bought my sixth Varminter in caliber 204 Ruger and for all-around, over-all Varminting useages I prefer the 204 Ruger!
Along with the slight edge in accuracy the 204 Ruger has LESS recoil, less barrel heat and (I am opining on this point!) less barrel wear than the 22-250 Remington.
Less recoil (muzzle jump/loss of sight picture) is VERY important to me when Hunting Varmints (including the two you mention).
The lethality of the 204 Ruger on Coyotes, Badgers, Rock Chucks and the like is just amazing to me.
I have killed Rock Chucks out to just past 400 yards with my 204's and IIRC my furthest Coyote kill with the 204 was just under 400 yards.
"What (which) of the two is better for coyotes and groundhogs" - in my experience the 204 Ruger is the correct answer.
Best of luck to you with whichever you choose.
Hold into the wind
VarmintGuy