What a fascinating question! I have included links below to the relevant enotes pages on both this text and the works of Charles Darwin, so hopefully they will help you to explore this question further. However, I suppose just to start you off, the most obvious comparison that can be made is the way in which both this text and the works of Charles Darwin try to "play God" in different ways. Let us remember that the work of Darwin, though mostly accepted as fact nowadays, was highly contraversial in his time, as it led to arguments that the world, and the creatures in it, were not created by God but had evolved and continue to evolve over time. This suggested either an absent God, that had created the earth like a watch to run smoothly and then could leave it to work by itself, or, even worse, suggested that there was in fact no God at all. Thus Darwin, in seeking to explore the laws of nature, transgressed the boundaries of acceptable scientific investigation.
In the same way, we see that Dr. Jekyll tries to "play God" and transgress those same boundaries of what is acceptable and what is not in terms of scientific enquiry by his attempt to explore and isolate the two contrasting sides that exist in humans. Note how he attempts to justify his experiments, talking about the isolation of the good part of humans from the bad:
if each ... could be housed in separate identities, life would be relieved of all that was unbearable; the unjust might go his way, delivered from the aspirations and remorse of his more upright twin; and the just could walk steadfastly and securely on his upward path, doing the good things in which he found his pleasure, and no longer exposed to disgrace and penitence by the hands of this extraneous evil. It was the curse of mankind that these incongruous faggots were thus bound together—that the agonized womb of consciousness, these polar twins should be continuously struggling.
However, as the rest of the story goes to show, Dr. Jekyll is dabbling in matters that are too complex and high and mighty for mere humans to understand, and he becomes a victim of his own scientific ambition, unleashing his own dark side and giving it dominance over himself.