PDIFF

Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them.

In summary, PDiff is more general than Diff because it allows pieces (corners), and one cannot in general smooth corners, while PL is no less general that PDiff because one can linearize pieces (more precisely, one may need to break them up into smaller pieces and then linearize, which is allowed in PDiff).

That every smooth (indeed, C1) manifold has a unique PL structure was originally proven in (Whitehead 1940).

The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in (Thurston 1997).

A very brief outline is given in (McMullen 1997), while a short but detailed proof is given in (Lurie 2009).

Splines are piecewise-smooth, hence in PDIFF, but not globally smooth or piecewise-linear, hence not in DIFF or PL.
PDIFF serves to relate DIFF and PL, and it is equivalent to PL.