If so, what causes this randomness?

Suppose there is no randomness, only causality. Then what causes the causality? The explanation for how the whole system works has to come from outside the system. That's.still true if there is no randomness.

Uncertainty would prevent identical copies to begin with.

And even if you could do that, which you can't, wave function collapse(s) would make them diverge almost instantly.

Chaos is a major factor. Small differences in initial conditions can grow to become large differences in final conditions.

If it's only "down to the atom" similar, then there's a lot of information not carried over into the other universe. The copy will diverge due to the different lower-level details even without considering annoying concepts such as those in quantum mechanics.

Quantum randomness is fundamentally random, unless you believe in hidden-variable theories, superdeterminism, or something something Bell's theorem loopholes.

This is true for both shut-up-and-calculate QM and for MWI; the difference is whether the universe is random, or whether the "branch" that your subjective experience ends up on is random. In the latter MWI case, I think any observer looking at the two clones Earths would still see divergence, because an observer is unable to somehow probe the universal wavefunction and see the deterministic evolution of wavefunctions or whatever anyway.

There's a separate question of whether you can "see" this randomness, and thus whether quantum randomness even matters. The answer is really yes. As one example, mutations can be caused by cosmic rays. Maybe an extremely fit genotype came to be because of a cosmic ray, that happened to come in the right direction due to the randomness from pion decay in the upper atmosphere. That would be a major macroscopic deviation that would happen over generations. There are probably many other such things. 

Also, quantum theory forbids wavefunction clones, so the initial state of your two Earths would already be different (different different or different Everettian branches). This is my understanding of the no-cloning theorem, at least.

I have a very limited understanding of this subject, but this is my best attempt at constructing a coherent answer: it's an open scientific/philosophical question.

Anything in scare-quotes ("Like this") below is a word or concept that I only have rudimentary, incomplete, or mistaken understanding of- you have been forewarned.

Currently, our best understanding of the time evolution of the universe is given by the standard model of particle physics, which is quantum mechanical and uses the Schrodinger wave equation to predict the behavior of subatomic particles like electrons. The Schrodinger wave equation gives PROBABILISTIC answers, and you only get "waveform collapse" and "the application of the Born rule" when you actually interact with the system ("make an observation" or "cause decoherence").

Now, there are some interpretations of this framework that basically say: that's it. The bedrock of reality involves random wavefunction collapse and you interpret that by handwaving or postulating Everett/Many-Worlds branches or whatever woo-woo you want. There are other interpretations of the framework that want to salvage pure determinism by insisting on unobservable "hidden variables" that secretly determine how a waveform will collapse, but a guy called John Bell created a thought experiment to test that and eventually someone actually ran the experiment for realsies and basically if there are hidden variables, they can't be "local" which I roughly understand as a fancy physics word that means 'dependent on spacetime coordinates'. Hidden variable theories totally work in non-local interpretations of QM (e.g. see "Bohmian pilot wave model"), but many physicist don't like those because it causes (even more) problems with General Relativity.

TL;DR / to sum up:

If there are no hidden variables in QM/QFT, then randomness is an intrinsic and foundational aspect of reality, so after "rewinding" the universe, differences will eventually accumulate due to differences in the way that the waveform of subatomic particles collapse.

I'd recommend reading Stephen Wolfram on this question. For instance: https://www.wolframscience.com/nks/p315--the-intrinsic-generation-of-randomness/

I understand that if you do this, quantum processes which happen constantly for every atom you cloned have random outcomes. If the universe is a simulation you could force the RNG seed to be the same and get the same outcome, but assuming you have to copy the earth as the "user" of this universe by some enormous equipment that prints all the atoms in about the same place, you can't do that.

You also have the issue that you couldn't print everything at once in the same instant so the copied earth will evolve with time different from the original., and also you probably have to destroy the original to make the copy. (Destructive scanning)

Also Heisenberg uncertainty means your atom printers are only approximate, there is a limit on how accurate they can be.

Note also that atom printers won't actually work by current understanding of chemistry, it's just a thought experiment.