I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a foundational subject to most mathematicians. A large proportion of low dimensional topologists use the foundational theorems in PL topology as black boxes, certainly without understanding or having read the proofs, and in fact they can do good mathematics that way. In the smooth category, the situation is even worse- I'm sure that there are very few people in the world who understand the proof of Kirby's Theorem, which is a difficult result, but it gets used all over low dimensional topology as a black box. Indeed, the fact that a diffeomorphism of $S^2$ extends to the $3$--ball is fundamental, under the hood everywhere, and highly non-trivial.
So you can be a manufacturer, or you can be a consumer. As a consumer, maybe you don't need to know PL topology beyond the basics that you need in order to understand simplicial homology and other basic constructions. A more sophisticated consumer might need more- I don't for example know a concrete smooth construction of linking pairings (the PL construction is in Schubert)- and in general, cell complexes allow you to work explicitly and concretely. PL proofs, if you read and care about proofs of fundamental results, tend to be shorter and easier than smooth proofs, which is not surprising because a-priori there is so much less structure which has to be carried around. This was indeed why Poincaré first considered triangulated manifolds; because of the technical facility which they afforded him. As a counter-point, I should point out Smale's comment in the introduction to in 1963 paper A survey of some recent developments in differential topology (which I recommend that you read, as it discusses your question):
It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet...
Another aspect, which is not to be sneezed at in today's world, is that PL manifolds are better suited to computers. This is indeed the focus of Matveev's book on "algorithmic topology".
Finally, as a PL question, I nominate:
Open problem: Construct a discrete $3$-dimensional Chern-Simons theory, compatible with gauge symmetry, replacing the path integrals of the smooth picture (which are not mathematically well-defined) with finite dimensional integrals.
