Pyramid Head's model does change between his sword and spear variant.
I knew I wasn't crazy. (Credit to the Silenthillmuseum model viewer site)

Occult(?) text appears on him. Wish Bloober kept the detail.
Beyond that nice detail, his helmet also changes with more white rust and parts of him get dirtier.

I believe the white rust appearing over the part where the tongue comes out indicates that it's sealed off now and serves as a visual reason for why he no longer has those animations with the spear. His helmet also slides open every time he impales Maria. So his helmet is able to change its own structure, it's as organic as he is (of course you already see and hear it creak and shift a little in the closet cutscene). Could also relate to how James changes from the first to second half.
Loudmouth Masahiro Ito should have explained this but I guess most fans gloss over it since it's hard to tell in-game. I am convinced it's meant to resemble text as if it's some scroll or passages and not just random cloth detail since the model already has grainy cloth details and the fact that it appears on him once he gets this ritualistic spear along with the extra white splotch which confirms it's intentional.

The grainy cloth fiber texture.

Faded text appears over it.

Most visible "passage" texture on his left side.
I imagine it as excerpts from the Crimson Ceremony. Matches with the "Red Pyramid Thing" descriptor.
This is the last post I'll make regarding his model since I think I've pretty much confirmed my suspicions upon deeper comparisons. Haven't seen anyone else talk about it and not much of a peep from Reddit or forums, so its nice to take away something a little new albeit purposeless given how sharp this fanbase usually is with detail.