What is the result that that differs by many orders of magnitude between QM and GR? It is well known that QM and GR are deemed incompatible due to a discrepancy in some calculations which I have read can differ by large magnitudes. What are these calculations to which people are referring i.e. what equations/results?
 A: You probably heard about the energy density of the vacuum. Quantum field theory seems to suggest a value about 120 orders of magnitude larger than value used in cosmological theories based on General Relativity. The low cosmological value is in good agreement with observation but it seems unnaturally low.
A: They are not incompatible just because of the discrepancy is some calculations. If that was true then Newton's law of universal gravitation would also be incompatible with GR. 
QM and GR are incompatible because they have fundamental differences. GR requires a smooth gravity while QM requires a quantized gravity; therefore, we are hunting for the graviton.
Unless you are referring to measured vs calculated difference of constant as mentioned by @Void, I am not sure what other discrepancies of large magnitude you are referring to.
A: Just want to echo the observation of @Void: Quantum gravity as an effective field theory has only insignificant impact on classical gravity (see, e.g. https://journals.aps.org/prd/abstract/10.1103/PhysRevD.50.3874). 
The problem lies in the quantum anything else other than quantum gravity: for instance, the quantum fluctuation of a typical bosonic field would have exorbitantly large contribution to the cosmological constant, contradicting the concordant cosmological model $\Lambda$CDM which is based on CLASSICAL gravity. 
In the high energy community, the savior is supersymmetry: the contributions to the vacuum energy from bosons and fermions as SUSY partners exactly cancels out, since they are of the opposite sign. However, there is a catch! SUSY has to be spontaneously broken to facilitate the Higgs mechanism, which entails an electroweak scale (still WAY too large) cosmological constant.
So is SUSY the true Messiah?
