Since the epistemological and methodological issues are similar with simulations, which are computer extended analogs of thought experiments, and computer assisted proofs I will only focus on the latter. Computer assisted proofs came into prominence with the Appel-Haken-Koch's (supposed) proof of the Four-Color Theorem. Tymoczko in his 1980 paper Computers, Proofs and Mathematicians wrote:"Computer-assisted proofs illustrate the need for a more realistic philosophy of mathematics that allows for fallibility and empirical elements". But this meant extending the scope of mathematics only while demarcating the "rigorous", computer-free one:
"The lemma claims that each configuration in a certain set is D-reducible (or C-reducible). The grounds for this lemma are that suitably programmed computers delivered certain output when given certain input. However, such grounds are inherently fallible. The output could have been misread, the computers might have malfunctioned, the computers might have been misprogrammed, the programs might not have captured the mathematical intention. If any of these possibilities obtained, then, for all we know, the Four-Color Theorem might be false... To be sure, the possibility for error is rather small. It does not preclude mathematicians from knowing the Four-Color Theorem any more than the possibility of error prevents scientists from knowing facts about the physical universe. But that small possibility of error does preclude mathematicians from knowing the Four-Color Theorem with absolute certainty. The proof is not rigorous."
Since then it was pointed out that "absolute certainty" was never absolute, and issues with social acceptance of proofs did not originate with computers. For instance, Kempe's 1879 proof of the four color theorem was accepted as such for 11 years, until Heawood found a flaw in it. A fierce debate about the role of rigor and experiments/simulations commenced when Jaffe and Quinn in 1993 suggested to implement Tymoczko's idea by demarcating "theoretical" (modeled on theoretical physics) and "rigorous" mathematics institutionally, in publications, etc. See discussion and references under What makes something mathematics? The institutionalization idea did not take (some saw the Jaffe-Quinn proposal as an attempt to devalue "theoretical mathematics"), but clear delineation of conjectural/heuristic material remains an informal norm. It is also expected that any substantive use of computer tools, even in calculations let alone proofs, is explicitly mentioned and described enough to be reproducible (which is similar to reporting experiments in science).
But hand-wringing over "absolute" rigor is not the only concern. Hamming, whose mathematics was computer science oriented, is well known for quipping "typing is no substitute for thinking" and "the purpose of computing is insight, not numbers". One can take it pragmatically, or one can take it more fundamentally. Even pragmatically, simulations of the hurricane movements, say, may give us predictions, but not why they are thus and so. When models disagree we are left in the dark, because the crucial element of insight is missing. And on the Kantian/intuitionistic conception of mathematics and mathematical validity the chasm between human and computer assisted proofs is more than a matter of degree. That a deduction constitutes a proof as long as its every step is according to the "rules" is a formalist idea. But not all scientists or mathematicians are formalists. To some there is that elusive aspect of intuitive insight without which a proof is at best a blind calculation, with a stigma of inferiority attached. "Understanding" just does not take place outside of an understander, and a proof is not a proof until it is understood. Similar sensibilities affect reception of quantum mechanics, complaints about "shut up and calculate" are not uncommon.
In Remarks on the Foundations of Mathematics, even before the advent of computers, Wittgenstein identified a general phenomenon that feeds such worries. To make a computer capable of assisting in a proof a lot of linguistic conversions are required. We typically assume "transparency of language" and see them as converting the "same" calculations into different formalisms, but supposed "conversions" may alter the conception of what is done. The "dangerous, deceptive thing" about it is that "it makes the determination of a concept - concept formation - look like a fact of nature":
"A mathematical proof must be perspicuous... The proof must be a configuration whose exact reproduction can be certain... What convinces us -- that is the proof: a configuration that does not convince us is not the proof, even when it can be shewn to exemplify the proved proposition. That means: it must not be necessary to make a physical investigation of the proof configuration in order to shew us what has been proved.
[...] I want to say: one need not acknowledge the Russellian technique of calculation at all -- and can prove by means of a different technique of calculation that there must be a Russellian proof of the proposition. But in that case, of course, the proposition is no longer based upon the Russellian proof.
[...] It is not logic - I should like to say--that compels me to accept a proposition... when there are a million variables in the first two pairs of brackets and two million in the third. I want to say: logic would not compel me to accept any proposition at all in this case. Something else compels me to accept such a proposition as in accord with logic... I want to say: with the logic of Principia Mathematica it would be possible to justify an arithmetic in which 1000 + 1 = 1000; and all that would be necessary for this purpose would be to doubt the sensible correctness of calculations. But if we do not doubt it, then it is not our conviction of the truth of logic that is responsible."
In other words, Wittgenstein rejects the formalism/implementation distinction (which implicitly appeals to language transcendent platonist entities "expressed" in different implementations), and points out that "conversion" creates a new formalism. Since in the case of computers one of them is "perspicuous" and the other not we have a qualitative difference between the two, not merely a pragmatic one. Of course, he would not object to the shift of accepting new proofs ("philosophy leaves everything as it is"), but rather to the philosophical self-deception that the leap taken here is nothing but a cosmetic makeover.
Such a surreptitious shift from constructive to platonist proofs is what motivated the intuitionist pushback, but as Wittgenstein points out the self-deception involved is far more pervasive. The point affects even formalists, as such shifts may involve major changes in the rules of the "formal games" of mathematics. On the other hand, it would be silly even for intuitionists not to make use of computer assisted proofs pragmatically. After all, when done by competent and trustworthy professionals the degree of certainty they provide is far greater than from numerical checks of conjectures, which have a venerable history involving Euler and Gauss.