Actually, it's not that complicated. The most obvious incompatibility is principle of superposition. Basically, Quantum Mechanics requires that equations that describe the system are linear. QM completely breaks down without this assumption. On the other hand, the field equations from General Relativity are non-linear.
Linearized gravity, which is basically Newtonian Gravity with added gravitomagnetic effects, can be quantized with some success. It's not terribly practical, but it makes some predictions about gravitational waves and gravitons.
Alternatively, you can do some limited quantum mechanics in GR setting, so long as effects of QM don't alter gravity too much. Hawking Radiation relies on this, for example.
So there are ways to make them work together in some limited situations, but overall, the equations are incompatible, and so a better theory is needed. My bet would be on QM being wrong and working only as a small-scale approximation, with true field theory being non-linear, same as that of gravity. But that's just a stab in the dark, really. Of course, we'll keep using QM in all likelihood regardless, due to it's great utility, same as we keep using Newtonian Mechanics despite its limitations.
