How is genetic engineering wrong, but cyborgs are okay?

Because Earth never faced an extinction-level Cyborg War, pretty much. I'm of the opinion that the primary reason for the Federation's ban on genetic engineering is Earth's enduring trauma from the Eugenics War.

Not a trekkie and writing from a real life perspective: genetic engineering cannot have the consent of the child whereas an adult may freely consent to become a cyborg. Moreover, genetic changes are permanent in the population whereas cyborg modifications end with the individual.

Those ideas coupled with the Federation having the legacy of the Eugenics Wars in the past probably motivate them to ban genetic engineering.

Star Trek celebrates the diversity of humanity. The extremes of genetic engineering and (on the other side of the spectrum, perhaps) the Borg are symbolic of the corruption of that diversity.

For an in-universe explanation, I suppose you could just look at the degree to which cybernetics are tolerated. Rutherford-level cybernetics? No problem! Borg Queen-level cybernetics? Helm, warp nine, full reverse!

Because old cultural taboos are not rational. Think of how for almost every animal species on earth there is a culture that labels it a delicacy and another culture that declares it impure and inedible.

As many have pointed out, trauma from the Eugenics wars is a big reason behind the Federations stance on genetic engineering, but that's just part of it. The Federation is a Utopian Meritocracy that celebrates diversity. Genetic Engineering undermines diversity and creates inequality in a society where everyone is granted an equal opportunity to rise and succeed. This is the other half of the equation as to why the UFP doesn't like genetic engineering.

The Federation's position on genetic engineering is not well considered

I have a theory that explains why genetic engineering and cybernetics are a rarity amongst the main galactic civilizations.

One can point at the Eugenics Wars as a reason for Earth, at least (and the Federation, as a result), to have a phobia of genetic engineering in particular and transhumanism in general. The lasting trauma gave humanity such a collective phobia that centuries later they almost totally forbid genetic modification, and seem to limit cybernetic augmentation to corrective prostheses like Geordi's VISOR. The encounter with the Borg reinforces this phobia, but its essence was already firmly in place.

The Eugenics Wars explain the Earth-originating phobia of transhumanism, but what about the rest of the galaxy? Vulcans might have their own half logical / half mystical reasons for rejecting augmentation, saying that such methods would be tantamount to trying to cheat one's way to achieving pure logic. Klingons might have lingering trauma from the augment plague of the Enterprise NX-01 era, combined with the belief that augmentation is dishonorable.

We don't know enough about the Gorn, the Tholians, the Sheliak, or the Breen to make informed guesses. They don't seem to be chock full of cybernetics, but as to why or why not, there isn't enough information.

But what about the Ferengi? The Romulans? The Cardassians?

All three have the means, motive, and opportunity to embrace either genetics or cybernetics, or both. None seem to be morally "above" the idea of using artificial means to advance themselves or get advantages over their enemies. And yet none seem to have embraced the idea. Why not?

This is where my theory comes in. I submit that every race that fully embraces transhumanism (Transbeingism? One doesn't want to be too Homo sapiens-centric.) eventually either self-destructs, is destroyed, or literally transcends our conventional understanding of civilization.

As for examples, look at the Bynars. They embraced cybernetics, and were one bad solar flare away from totally destruction, saved only by a desperate theft of the Enterprise to reboot their entire planet, dependent on the good will of the victims of their theft. Look at Earth, where experimenting with genetic augmentation nearly destroyed humanity. Consider the children of Landru, who were arguably under the control of a fully cyberized intelligence, and were totally stagnant, and well on their way to extinction.

Next, consider the Borg, a fully cyberized species that has taken on the nature of an uncontrolled cancer, gobbling up unique life and technology. If they continue as they have, the other powers of the galaxy will have no choice but to destroy them out of self-defense. Also look at the example of Gary Mitchell, augmented by the Galactic Barrier, who went mad and had to be killed by Kirk before he could pose a greater threat.

Finally, consider the many beings we have seen who seem to have ascended beyond mere mortality, to the degree that they barely interact with the universe any more. Trelane, and his parents. Kevin Uxbridge. The Traveller. At the very upper scale, possibly the end result of all of this upward mobility, are the Q Continuum. But all of these beings exist in realms that defy our understanding of technology and biology. These are beings for whom cybernetics and genetic augmentation have long since been left behind; they have escaped destruction, but they also cannot remain in the same community of mere mortal beings as most other sentients in the galaxy.

In essence, augmentation is one of the great filters of galactic civilization. When a species begins tampering with its own existence, more often than not it is destroyed, but those that are not destroyed must continue to ascend; they cannot remain in a form that we would recognize as mundane life, only moreso.

TLDR: You either die the Bynars, or augment long enough to join the Q.

It isn't won't it's a tool and any tool can be used in good ways or bad ways. It's society that needs to figure out what is good or bad.