They're different issues but NOT contradictory. The reason why one is for small scale and one for large is not that they are doing same thing and working at different scales, but that observations are possible on the small or large scale. They're really doing different things though.
Quantum theory says energy is quantized (i.e. digital "steps" of energy levels, the perfect example is emission of certain spectrums of light from electron transitions in atoms). So the electron has different energy levels. So does a baseball. It's just that the levels are SOOOO freaking close together at that scale, that you can't measure different speeds of a baseball and see the steps.
Relativity says that energy has mass associated with it. So if I speed up a baseball, it actually gets heavier. (This is why you can't make a rocket go faster than the speed of light...it gets heavier and heavier and you can only approach the speed of light asymptotically.) But of course at the baseball level, this is irrelevant. (It occurs, just the effect is so small it's irrelevant and too small too measure). However for speeds with some reasonable fraction of the speed of light, you can notice it.
This is what Michelson showed with some of his speed of light measurements (it was a constant, rather than "adding" or "subtracting" the speed of the Earth as it moves). He didn't understand the effect, but he was too much of a good scientist to ignore it:
Now: In general, we "need" quantum mechanics at small scales. (It's both that we need it do do proper math calcs of effects and also that you can really only observe it with instruments at these scales...it's sort of the same issue.) Similarly for relativity at large scales. You don't need EITHER ONE for the baseball. Not only don't you need it (to do practical calcs on the baseball), you really can't observe either effect at the speed and mass involved in a baseball going ~100 mph during a pitch. So the classical Newtonian theory is much more convenient to use. It's like their is zero point in using a telescopic sight on a shotgun. But either theory would still "work" on the baseball if you wanted to bother (the relativistic math is probably a little bit less of a pain than the QM...easier to see it collapse to the classical...QM math is a hassle even at small scales, but you don't have any choice when you need it.)
Note that relativistic effects DO occur in small scales. The perfect example is the electrons of heavy elements. Like the electrons of a heavy element like lead or uranium. Those little buggers are flying! So, it turns out you "need" relativistic QM for doing quantum calculations involving chemistry of very heavy elements. (You may need it for even smaller elements to observe very fine effects, but you totally need it to get even reasonable calculations of orbitals of heavy elements). A classic example of a relativistic QM effect is "why is gold yellow". See here:
Note that the relativistic QM equation dates back to the 20s (Pauli and Dirac). It is a little bit more of a hassle to deal with than the simpler non-relativistic version, so for some problems people don't bother. However, it also has some tangential benefits in dealing with a property called electron "spin", regardless of relativistic speeds. But there is NOT some massive problem of resolving the two effects (energy quantization and relativistic mass). It was done a few years after Schroedinger, all the way back in the 20s.
https://en.wikipedia.org/wiki/Relativistic_quantum_mechanics#History (this article is a little hairier than the others...what can I say, math/physics geeks on wiki sometimes lose the plot in terms of teaching laymen versus using articles to explain things to themselves. But the section I linked "history" gives a bit of the story, I describe above.
Caveat: both QM and relativity are complicated phenomena and I have not discussed all aspects (e.g. uncertainty principle, tunneling, special versus general relativity, etc.) Nor do I have perfect math or physical understanding of them. But I think this answer fundamentally contradicts some of the other discussion here that looks at the two as disparate theories that don't work with each other. They do.