This has thrown me a bit. I will readily admit that I am very far from having a great understanding. More a fascination.
But.
As I understood it, they are each applicable to different states. Classical is still correct for the everyday things but breaks down at the very small. And this is where Quantum takes over.
The reason I understand this is because the laws of one don't apply to the other. And that is why science had been striving to come up with a theory which brings them both together.
Classical mechanics is just an approximation, and in a fundamental level, it is totally wrong. For example, having a computer with infinite power, and knowing the position and the momentum of each particle in the universe, by using classical mechanics, you can predict everything that is going to happen for eternity. In essence, it is a deterministic theory.
Quantum mechanics says that this is not the case. The universe is ruled by quantum laws, which are probabilistic in nature (Schrodinger's equation is not probabilistic, but the wave-state of particles it is). Predicting when a wave is going to collapse is impossible, though you can make probabilistic predictions for it. So here come two problems: 1) it is impossible to know both the position and the momentum of a particle, in fact, the more precisely you know one, the less precise you are in the other (Heisenberg uncertainty principle); 2) even if that was the case and you know that, and have such a computer, still you won't be able to predict the future. Essentially, the universe is not deterministic.
We are totally sure that QM is either correct or mostly correct (it could be an approximation of something ever deeper), and we know that classical mechanics is wrong. It is not that they contradict each other, more that one is correct and the other isn't. Just that classical mechanics is very simple and is a very good approximation of quantum mechanics when it comes to everyday's life and large objects (by large, I mean everything bigger than molecules), but it is fundamentally wrong. Similar to how Newtonian gravity is wrong, and it is just an approximation of general relativity (which most likely is a good approximation of a yet-to-be-discovered theory of quantum gravity).
The problem in physics is how to unify quantum mechanics and gravity. Quantum mechanics describe all the particles and all the forces except gravity. But gravity is very important especially for large objects, and so far, the best theory of it is Einstein's general relativity. For the most part, as long as you use gravity to study large objects, you are fine (the reason being that quantum effects in large objects are so small that they can be ignored). However, when you go to tiny objects with massive mass (black holes), everything breaks down. You cannot ignore quantum mechanical effects, and suddenly the equations start showing infinities everywhere. This is why people say that quantum mechanics and general relativity do not agree with each other (on the other hand, quantum mechanics is totally fine with special relativity). And considering that everything is quantum, people think that gravity must be quantum too, so general relativity is probably just a very good approximation.
The attempts to unite these two fields, have not succeeded so far. String theory unifies them in a very elegant way, and actually, gravity emerges from it, so you do not need to even add it. Unfortunately, there is no experimental evidence that string theory is anything more than just a correct mathematical theory. It might just describe a hypothetical multiverse, but our universe might not be part of that, so at this stage, it is just math. Quantum loop gravity has also tried to have a theory of quantum gravity, but also there the experimental evidence is small, and unlike string theory, it is not a theory of everything. I think that there are other fields that try to unify QM and GR, but for the most part, strong theory has been the one where almost everyone is working (and everyone else hates it).