I like the 22-250 & I have owned 3 of them & a Swift that was great, but not relevant here.
We should not look at these 2 in a stand alone situation as though these were all you had, in that case
the criteria would be totally different.
No, we should look at what each cartridge brings to the table, in light of the barrels you already have.
The 204 & the 243 will do everything that you would do with a 22-250, but do it better on both ends of that spectrum.
For small Varmits (Groundhogs & down), the 204 will do as good a job with a very similar trajectory. For PD's or other volume varmit shooting the 204 offers the additional advantages of using much less powder, being able to spot your own shots if your gun has a decent weight & a much better barrel life & the tube not heating up so fast. You will burn alot less powder as well & beats the 22-250 bad for these chores. This handles the lower end.
For big Varmits, windy day shooting & any big game hunting the 243 is much better with again simular trajectory but much less wind drift. If you decide to shoot way out there, the 243 will be more decisive & the BC values are far better than 22 cal bullets. For Deer size game, you have energy for a little longer shots & better/more variety of game bullets. And this more than handles the upper end.
The 22-250 offers nothing that you don't already have & more. But the 223 offers alot!! With plentiful & more economical brass & a little less powder than a 204 & a BUNCH less powder than the 22-250, it
is by far the cheapest to shoot. It will do about any Varmit chore you wish under 300 yards. It is extremely accurate & more consistant with the Handi platform than most. Everyone I know that really shoots alot WILL have a 223 in his arsenal. Go to
www.6mmbr.com & check out the 223 article.