Given a uniform hypergraph on vertex set $V$, construct a simple graph on $V$ by joining $v, w\in V$ whenever they are contained in some hyperedge in $E$.
Conversely, given a graph on $V$ that admits a (not necessarily disjoint) edge decomposition into copies of $K_r$, set $h\in V^{(r)}$ to be a hyperedge whenever the $G$-induced subgraph on $h$ is complete.
As far as I see, these two constructs capture all information about one another and theorems about one can be translated to theorems about the other.
If this is true, what then is the additional value in considering the concept of a uniform hypergraph? Is there some intrinsic difference useful in a certain kind of problems or is this just because the hypergraph structure is more natural (if it is at all, seeing as we could equally well just consider a graph along with "generalized edges" given by its complete subgraphs...)?