When Marvel created the Ultimate Universe, it was designed to be a world for a contemporary audience that could reflect the world currently lived in and be more accessible than the long history of Earth-616. This meant sweeping changes for many of Marvel's heroes and villains. Omega Red in particular has a few key features that differentiate from his mainstream counterpart. For starters, he is not an enemy of the X-Men, but rather Spider-Man, and only through extraordinarily bad timing. However, there were a few aspects of his character that kept him in line with his main universe counterpart.
The Arkady Rossovich of Earth-1610 debuted in Ultimate Spider-Man #86 by Brian Michael Bendis and Mark Bagley. He was a mutant with a hatred for humanity who volunteered for experimentation while in prison in his home country of Russia. This Omega Red became a hitman, taking high-profile jobs from wealthy employers who wanted to see their enemies dead. His choice of profession was what ultimately put him on a collision course with Spider-Man that ruined his career.
After a job to assassinate the head of the Roxxon Corporation was foiled by Spider-Man, Omega Red was taken into custody by S.H.I.E.L.D. and from there his fortunes sank. An article written by J. Jonah Jameson detailing the hitman's humiliation at the hands of Spider-Man severely damaged his reputation, making him seem incompetent and unreliable as a hired gun. All of Omega Red's clients turned on him, leaving him without a source of income.
Enraged, Omega Red escaped and infiltrated the Daily Bugle to try and murder Jameson. Once again, Spider-Man swung into action and knocked Omega Red out with a vending machine, depriving him of his mark and almost ensuring that Omega Red's career would never recover. But really, it was pure happenstance that he became Spider-Man's enemy, a classic case of being in the wrong place at the wrong time.
This set him apart from his Earth-616 counterpart where his usual enemies are the X-Men or more specifically, Wolverine. But he never encountered them throughout Earth-1610's existence. Even so, there were still several similarities to his main self. His powers were very much the same, being able to use tentacles that extend from his body as offensive weapons. And he is still a mutant, so the origins of his abilities remained unchanged.
But perhaps the single most important trait that he shares with the main Omega Red is that they are both relentless killers. The origin of Arkady Rossovich in the main Marvel Universe has gone through revisions over the years, but they usually follow the same path. Arkady starts off as a sadistic serial killer or murderer of some sort, often being described as pure evil. And it seems that this Omega Red has a similar taste for snuffing out the lives of others.
Although it is established that he has a hatred of humans, it is his choice of career that brings the point home. Omega Red chose to use his powers to become a hired killer. The service he offered was death and usually against human targets. So not only did he satisfy his hatred, but he got paid to do something he loved.
However, unlike the Omega Red of Earth-616, this one couldn't maintain his intimidating persona. Repeated humiliations by Spider-Man, then a mere teenager, meant that the once fearsome killer was reduced to someone who proved unable to handle a child in the eyes of his clients. And even after his world's Peter Parker died, Omega Red was beaten by Miles Morales when he first became Spider-Man, someone even younger than Peter was at the time of his first defeat.