How can a pair of eldritch twins be polar opposites despite their shared heritage?

[Erebus]

An secret and powerful cult, known as the Order of Dunwich, devised a millennia-long eugenics program for the purpose of incarnating an ancient deity into our reality. Over thousands of years, they would practice selective breeding among their members, recruiting outsiders when necessary or obtaining genetic samples through other means. The end goal is to eventually create the perfect mother, in order to give birth to the perfect host for this God to walk the earth.

Many generations of this has led to an albino woman found to be compatible to bear an eldritch God, named Lavina Whateley. However, a complication was thrown into the plan, as she unexpectedly produced two sons. One, named Wilbur Whateley, appears as a perfectly normal child on the outside despite it's eldritch heritage. The other appears as a monstrous creature simply referred to as the Dunwich horror. As it is unable to hide it's true nature, it is widely regarded as a failure.

How could these twins, despite having the same genetics, appear completely different?

 

[Queshire]

Easy answer is have them be fraternal twins. Other easy answer is magic.

Less easy answer is that Wilbur's a jerk who prevented Dunwich from developing properly somehow in the womb (Wilbur's umbillical cord ending up wrapped around Dunwhich's neck maybe? Meh, I'm not a doctor) resulting in physical deformity upon birth which was compounded by being seen as a monster by the order as he was growing resulting in him getting poorer food, living in worse conditions than his brother, scars and poorly healed broken bones from abuse, etc, etc and so on, resulting in them developing in notably different manners as they grow.

 

[Spacebar]

Not all deformities are genetic. Some are developmental. Fetal Alcohol Syndrome, for example, is caused by a pregnant woman consuming alcohol during gestation, and results in both physical deformities and mental impairment. A perfect example for your case would be Autism, where one identical twin can develop it and not the other. The cause for this, as with the cause for Autism in general, is not known to modern medicine. So unless it's important for your theme or setting, it's not unreasonable for your characters to have no explanation for the Horror's deformity. With a sample size of one, it would actually be weirder if they did know.

But if you really want to get into it, then another question you're going to have to answer is: why has no precedent for this Horror deformity occurred in the past thousand years of breeding experiments?

 

[Prince of Spires]

because magic.

Because this woman had twins, the eldritch god was split in half, with both kids getting half of his abilities / looks. The good looking one simply got that part of the gods looks. The ugly one got something else.

I doubt most readers will wonder or care beyond that.

If you want a genetics answer. They're fraternal twins. In that case, there's nothing that makes these two childres the same. They are just siblings in the same sense that my sister and me are (except that they share a birthdate). There's no one questioning why my sister is femal and I'm male. Same parents, same genetic origine, but very different people.