There is a construction given in Theorem II.2.2.10 of the Handbook of
Combinatorial designs for a STS of order a prime of form $6t+1$. It is clear
that the resulting STS has a cyclic automorphism acting fixed-point-freely on
points. The other STS of order 13 has a full automorphism group isomorphic to $S_{3}$.
So the automorphism group could be a good invariant, depending on what you mean
by simple.

Note that it is possible to label the blocks of the designs so that they differ
in just six blocks (they differ by a switching operation). So it seems unlikely that a simple test which looks at local properties of an STS will be able to distinguish the two triple systems, though some less computationally expensive invariant may suffice in this small case.