We're not so much looking to replace the Standard Model (it works very well), but rather trying to figure out if there's something more, and preferably less complicated, under the hood. There are lots of hints that there's something more, and everyone is looking hard in lots of places to discover the next scrap of the treasure map.

If the tau were indeed found to exhibit substantially different decay properties from the muon, it would be major news within fundamental physics. The stated statistical significance of the result is quite low, you'll want to wait for more data (which are imminent at LHCb).

A different experiment, BaBar, also sees a similar excess, and the experiments roughly agree. Taken together, the results are near the threshold for "evidence for", but not "discovery of" a new result.

One upside to experiments with leptons (electrons, muons, and taus) is that they're simpler to understand. Anything involving hadronic physics (quarks, gluons, and the zoo of particles made from them, like protons and neutrons) is comparatively difficult to understand.

TL;DR: Stay tuned. The result is of potential importance, but not yet, and could fluctuate away.