I would choose 200 over 300 any day and I have both. 200 is just a "special" lens and 1 full stop faster than the other one which makes a significant difference (6400 vs 12800 or 12800 vs 25600 ISO). There is also way too much compression with 300 for it to be a good portrait lens (or let's just as just as good as the 200). 200, to me, is "the" head/shoulder portrait lens. It's bokeh is also unbelievably good, I would say the best lens in Canon's line up for portraits.
300 is just a "reach" lens and mostly used for birding or air shows (with a 2x extender), nothing more than that. I always put the 200 on whenever I feel creative and I want a magical look whereas 300 is a "tool" that gets the job done.
Also, both lenses take 2x III very well. That being said, as with all lenses, they need to be stopped down. 200 has a big advantage there because its a full stop faster. When you stop it down to F5.6, it's already sharper than 300. And as the aperture gets smaller, the light makes even more difference.
I hope that clears it.