Do you know any good places I can get a fairly simple description of the long answer? (I.e. a good place to learn about why the interface between GR and QM is problematic)?
That said, I can illustrate the conflict. The fundamental idea of general relativity is that mass warps the structure of space-time, and we perceive this warpage as gravity. One fundamental idea of quantum mechanics is that a quantum mechanical system is always in some superposition of possible states, and the superposition adds. (Since we're dealing with a complex wave, sometimes adding 2 possibilities leads to cancellation in one place and strengthening in another, creating interference patterns. But fundamentally we're just adding waves.)
One of the things that nobody knows how to do is add different kinds of warpage of space-time in a linear way and get a sensible answer.
Let's try something else. In QM perturbations of any kind of field must have an associated particle with describable characteristics. For a random instance vibrations in a lattice of atoms can be described by a field that must have an associated particle. That particle is called the http://en.wikipedia.org/wiki/Phonon. (Phonons are not fundamental physical particles, but they do exist mathematically and their effects have been experimentally measured.)
Now in QM the exchange of particles actually gives rise to various forces. Furthermore the ability for virtual particles to arise and disappear as long as you stay under the Heisenberg limit causes the prospect of particles interacting with various things, including themselves. When you try to calculate all of these interactions you wind up getting infinities added and subtracted with each other. There is a mathematical trick called renormalization that lets you cancel these out.
Now in General Relativity you have gravitational waves. (They have never been directly detected, but binary star systems have been observed losing power in accord with the predictions of GR.) If you try to fit these waves into QM you should wind up with particles called gravitons. But when you try to do renormalization on them, the procedure fails and you get infinities out.
So we have failures both ways. Attempts to state QM within the framework of GR fails because we don't know how to add perturbations in coordinate systems. Fields that GR predicts exist cause QM to break down and give impossible answers. The theories just don't merge. Experiment doesn't help because each theory is a very good description in its own region, and we can't get data on how the theories break down between their areas of specialization because we can't create regions with high gravitational curvature on a scale where quantum mechanical effects matter.
If you want to know more about some of the attempts to reconcile the theories I'd suggest reading up on "string theory", "quantum loop gravity", and trying to understand John Baez. I'm not a physicist and don't understand John Baez myself, so I can only wish you good luck if you choose to dive more deeply into this.