I'll convert my comment to an answer:
Yes, triangulations can distinguish two non-diffeomorphic smooth structures on any 4-dimensional manifold; in particular, given an exotic $RP^4$, there exists an exotic triangulation of topological $RP^4$ which is not PL-isomorphic to the standard triangulation. The reason is 2-fold:
a. The easy part is that each smooth manifold $(M, s)$ (regardless of its dimension) admits a compatible PL structure: One can find a smooth triangulation $\tau_s$ of $M$ whose links will be triangulated spheres.
b. The hard part is a theorem due to Kirby and Siebenmann,
Kirby, Robion C.; Siebenmann, Laurence C., Foundational essays on topological manifolds, smoothings and triangulations, Annals of Mathematics Studies, 88. Princeton, N.J.: Princeton University Press and University of Tokyo Press. V, 355 p. hbk: $ 24.50; pbk: $ 10.75 (1977). ZBL0361.57004.
that in dimensions $\le 6$, the categories PL and DIFF are equivalent.
In particular, if $s_1, s_2$ are non-diffeomorphic smooth structures on a topological manifold $M$ of dimension $\le 6$, then $\tau_i=\tau_{s_i}, i=1,2$, define non-isomorphic PL structures on $M$. Concretely, one can say that triangulations given by $\tau_1, \tau_2$ do not admit isomorphic subdivisions. (This property fails in dimension 7: Famously, there are 28 non-diffeomorphic smooth structures on $S^7$, but all PL structures on $S^7$ are PL-isomorphic. The other difference between DIFF and PL categories in dimensions $\ge 7$ is that there are PL manifolds of dimension $\ge 7$ which do not admit compatible smooth structures.)
Here one is working with unordered simplicial complexes. Thus, "branching structures" which one can assign (possibly after a subdivision) to triangulations $\tau_1, \tau_2$ are irrelevant.