As mentioned above both the calibres you have mentioned are good for varminting. Both have pro's and cons.
In an ideal world I would get a 204 over a 223 if you are varminting. From what I have seen, they are not the barrel burners that the 22/250s are.
With Varminting you get longer ranges on small critters easier with smaller flatter shooting projectiles and the 204 does this very well with a very small 30grain projectile going down range very fast. I have seen guys hit magpies at 350m or more.
The 223 is a bit moe versatile but again there are trade offs.
The 1:7 or 1:8 twist ones work well with the heavier ranges of projectiles such as 55gr, 62gr right up to 75gr, but for varminting where even the humble 22LR punches clean through your game, you dont need to retain energy with the heavier bullet and the lighter 35gr and 40gr ones are brilliant. But these really need the slower 1:12 twist to get the best acuracy.
For rabbits etc at 200+ metres you need a decent scope. I would not go below a 4-16 and would actually suggest 6-24 would be better. A rabbit at 300m is a very small target in a 4-12. And with a proper varmint rig there is no reason that you wont get out to 300m.
Finally, varminting rigs tend to be heavier for a more stable platform and repeat accuracy after multiple shots so are not condusive to lots of carrying so if you want a gun to chase pigs, get a bigger calibre suited to the task rather than compromise on your varminter.
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