This is a very vague question.

We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain complexes are modules of dual numbers $\mathbb{R}^{1|1}$(maybe not so correct, the multiplication here differs from the usual one), more generally, modules of a operad T (Lie algebra and associative algebra are of this kind). To get the infinitified version(L-infinity algebra, A-infinity algebra, etc.) one uses an infinity operad. Note that module is basically a morphism from T to some Auto( ). And the infinitified version basically replace the source thing by a higher thing.

Projective representation goes differently. A representation is the same as a module, a morphism from T to some Auto( ). Projective representation (we could generalize this idea, such as a group representation in a 2-category) basically replaces the target by a higher thing.

A nature question: whether these two constructions are the same? It seems obvious not. On the other hand, I feel they should have some relations.