I feel sorry for anyone who watches the remake before the original, spoils what is one of the great mainstream horror films.
A lot of the scenes and dialogue appear as they did in the original. Hence why David Seltzer (writer of the 1976 Omen screenplay) is credited as writer of the remake even though he wasn't involved at all.
The beginning is different, set in the Vatican and controversially using footage of real-life disasters (such as 9/11) to signal the impending arrival of the anti-christ.
Other sequences have been changed, generally to no good effect. The famous outdoor baboon sequence was swapped for a blander indoors version. The spectacular death of Lee Remick (Thorn's wife) was also made much flatter. The decapition scene was changed from a sheet of glass to a sign.
One of the other changes was to signal much more clearly that Damien was malevolent and knew he was all along. In the original, it was suggested all the deaths could be accidents that Damien was unaware of, until that great moment when he smiles into the camera

By making Damien so obviously evil, director John Moore completely misjudged the whole thing. The film is littered with similar mistakes, Julia Stiles is one of the worst pieces of miscasting in film history. But then given that Moore also directed a similarly pointless remake of "Flight of the Phoenix", it's no surprise it turned out to be a dud.