These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
When exactly does Damien truly become evil and aware of what he is? In the sequel he is shocked to discover his true destiny but many of his actions prior to then hinted he was aware of it at some level or at least knew he was different. There are viewers that believe he was actively evil from the outset, and simply unaware of why, believing he knowingly attempted to kill his own mother as a child for instance. Some viewers feel he is actually not directly evil but rather unwittingly channels it. And is in fact an otherwise normal child until he embraces his destiny. This would mean they see Mark being his first direct kill, and his true Start of Darkness.
Upon rewatching the original movie, Robert Thorn is a cold and distant father, which can ask two questions: is he this way towards Damien because Damien is a naturally unlovable child (just by nature of being the Antichrist), or is it because of the relationship between the father and the son, that Damien is going to be more inclined towards becoming evil.
Complete Monster: The Omen series has The Antichrist himself, Damien Thorn. What's interesting about him is that at first, he was horrified when he discovered that he was the Antichrist, but as he grew older, he came to terms with his destiny, and set out to do his father's wishes. In the second film, he revealed to his cousin Mark that he was the son of the Devil, and asked him to join him in his attempt at world conquest. When Mark refused his offer, Damien murders him by introducing an aneurysm into his brain and later set his aunt and uncle on fire at the end of the film. In the final Omen film, The Final Conflict, we are introduced to a grown up Damien, and we also discover that he became the Ambassador of Great Britain. At this point in his life, he had fully come to terms with his role as the Antichrist, and any redeeming factors that he had in the previous films had vanished at this point in time. When he realized that the Second Coming was on the horizon, he makes a decree that every baby boy in England that was born on March 24, 1982, be sentenced to death as a means of stopping the Christ child from being reborn. Along the way, he gets romantically involved with a female journalist, whom he then rapes, and he also manipulated her son into following him. He later uses said son as a Human Shield when Father De Carlo attempted to use the Dagger of Megiddo against him. Damien then tries to strangle the priest to death. Damien also manipulated Barbara into killing her husband with an iron, and she unintentionally helps Damien in his attempt to kill the Christ child by killing her baby.
It's the Same, Now It Sucks: Critics were quick to point out the 2006 remake whilst far from terrible was basically a scene for scene remake and didn't really do anything the original hadn't. The only significant changes story-wise being the time period, a few weird nightmare sequences, and the murder of Katherine.
Moral Event Horizon: Whilst Damien is seen by many as inherently and knowingly evil, thus crossing the horizon the instance he drew breath, the moment he kills Mark is pretty much the death of what little shred of humanity he possessed. In the remake he's more overtly evil from the outset helping to kill off his mother is the point of no return.
The CGI heavy version of these death scenes in the 2006 remake are arguably equally silly without the excuse of being over three decades old.
The nightmares/visions the Thorns have. From the smiling Nosferatu guy, who goes on to drop what is obviously a blood covered doll, to Damien wearing what looks like a grey version of the mask Jim Carrey wore during a certain film back in the 90's.
They Changed It, Now It Sucks: The 2006 score doesn't use the original (not to mention iconic) Jerry Goldsmith score. The new score is very understated in comparison and for some critics fails to leave any noticeable impact at all.
What Could Have Been: William Holden turned down the role of Robert Thorn. Seeing as how the finished film turned out, he jumped at the chance to star in the sequel.
Charlton Heston chose to do Midway instead, allowing Gregory Peck to come out of retirement to play Thorn.
Richard Donner could have directed The Final Conflict except he was involved in a dispute with the Salkinds over his firing from Superman II.
Marlon Brando, Gene Hackman, and Jack Nicholson were considered to play the grown-up Damien, until the producers went for the younger (then unknown) Sam Neill.