In vim, what is the difference between <c-r>+register and <c-r><c-p>+register with the . command?

I struggle to understand the solution to this vimgolf challenge.

The best proposed solution is

cw(<C-R><C-P>")<Esc>w.w.ZZ

Then I tried to do

cw(<C-R>")<Esc>w.w.ZZ

But then the text becomes

(one) (one)
(one)

instead of

(one) (two)
(three)

Can someone help me to understand why . behaves differently with those two commands?