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Here's my recollection from studying it 20 years ago.
The difference between classical and quantum physics is that classical is deterministic. Given starting conditions and enough computing power, once you specify the initial conditions of a classical system, its evolution is entirely determined (this is even true of chaotic classical systems).
In quantum physics, outcomes are only probabilistic. This is a result of (a) superimposition of complex wavefunctions (Schrodinger formalism) or equivalently (b) non-commutativity of operators (Heisenberg formalism).
It is often not appreciated that relativity is strictly in the classical physics category (ie deterministic).
As to conflict, yes as mentioned below in a sense there is none - it's just that quantum extends classical to regimes in which classical breaks down. However in another sense there is. Physics currently models the universe with two 'big' theories, quantum mechanics and general relativity. Each works wonderfully well in its range of applicability, however you wind up with problems where they both should apply (big bang, inside black holes).
The simple version is that GR always treats things as having definite values (including position) where QM says that plenty of things generally can't have definite values (position being one). The full version is a blizzard of maths.
Trying to extend QM to encompass GR (gravitons etc) is not something anyone has ever really managed to do in a mathematically coherent and testable way. String theory has some promise, but suffers from lack of testability given current equipment, but also the more serious problem that it looks suspiciously like an attempt to simply chuck more and more parameters at the problem so that, of course, you'll get a fit eventually, though not one that tells you anything physical.
In short, it's a real headscratcher.
Always happy to hear a contrary view.
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