Private stock won't make any differences in how a CU24 compares to a McCarty, except that both will be a little slicker and better at what they are than Core. The McCarty still has the Pattern neck, the CU24 you're looking at still has the Pattern Regular, the switching systems are different, and the pickups and body thicknesses are different.
There are differences in feel due to the neck carves and the number of frets. There are differences in tone due to the placement of the pickups being different, and of course, the pickups themselves are different as are the electronics.
But everyone's hands and fingers are different, so in terms of playing differences, that's going to be a very individual thing and impossible to generalize.
Pickups are obvious - the 59/09s are a little hotter and brighter than 57/08s. A blade switch operates differently from a toggle switch with a pull-coil tap. The CU24 will have a more focused tone, the McCarty a more vintage tone.
The ebony fretboard will be a little bit crisper on the attack than the rosewood one, but the rosewood might be warmer and sweeter. Everything depends on the individual instrument.
But you asked about playing differences, and that can mean a lot of things.
So some details as to what you're interested in will help. Or just go play some CU24s and McCartys and see for yourself.