Two great online/catalog off-the-rack shirtmakers. I've gone through over a dozen PF shirts in my day and have another five or six in the closet. I recently took advantage of CT's 4-for-$150 deal for new customers. Thoughts:
Material: When it comes to patterned materials, CT tends to have an edge. The fabric feels more substantial. However, their white oxford is actually a bit less substantial than PF's although also somewhat softer. It definitely feels nicer on the skin but doesn't necessarily look more luxe until you get up close.
Neck fit: I wear a 15.5 neck, and PF fits better out of the box without a doubt. CT is a little baggy at this point. However, after enough dry-cleanings, the PF collars shrink to the point that they're no longer comfortable; I assume the CT shirts will do the same after some use and fit well until they no longer fit.
Arm fit: I have skinny wrists, and that means PF's shirts (whether barrel or French cuff) tend to swallow my hands, especially out of the box before the dry-cleaning process has shrunk them a bit. As if this were possible, CT's sleevehole is even larger than PF's, so I find myself twisting my wrists a bit so that the widest part of the sleevehole is at an angle to my hand and doesn't fall as far down. Be interesting to see whether the shirts wear out before dry cleaning shrinks the cuffs enough. This problem, more than any other, makes me consider made-to-measure shirts and the attendant expense.
CT shirts don't have a gauntlet button, but the gauntlet opening is a lot shorter, so it doesn't seem to cause any problems. Also, CT mostly only offers sleeve sizes in odd numbers, so if you're a 34 sleeve like me, you have to pay extra per shirt for alterations to get the desired length.
Waist fit: CT offers an Extra Slim fit that PF just doesn't match, and it's worth it--PF's Trim Fit still has an extra several inches of material that I don't need, whereas CT has perhaps 1/2" extra. PF's skirts are a little longer, which is nice, but I haven't had a CT shirt come untucked on me, either.
Price: CT is between 1.5 and 2 times more expensive than PF.
For now, I'm not sure whether the price premium is worth it, though I suspect that it is. Once I see how the CT shirts age, I will have a better answer.