When Ridley Scott made Prometheus, he shoved away the continuity established 8 years prior in the first AvP movie. Although it can be said that AvP was never canon, it had some interesting ideas about the origins of the Aliens and the Weyland-Yutani corporation. I am quite sure Ridley never saw AvP, but one of the scriptwriters for Prometheus might have. Here are the main similarities that I found.
Both movies feature an aging and dying Weyland on a quest for longer life. Both Weylands are the founders of the Weyland Corporation and have advanced robotics in their past. Both Weylands die in a similar way during the 3rd act of the movie. Lance Hendriksen has the benefit of not wearing bad makeup and being an actually likable character.
The mission picks up as a signal is received from a far away place and a team is assembled. The briefing for the mission is given on a ship (or a spaceship in Prometheus) by Weyland. Many characters have no idea for what they signed up for before this.
Both movies feature a scene in the beginning where archeologists are excavating on a dig site on Earth. Some of the main characters of the movie are established in this scene (Sebastian in AvP, Shaw and Halloway in Prometheus).
Both movies establish that an advanced civilization with big spaceships has been on Earth before and helped humans "evolve". The engineers used their DNA to spawn humans and the Predators used humans to breed the Aliens. The signs of this advanced civilization have been mostly wiped out. The predator technology is still evident by the pyramids and signs of the engineers are found in a cave.
Both movies feature a team of a dozen people including scientists and mercenaries. The mercenaries are one the first to be killed off. The lead character is a young woman who has a romantic relationship with another scientist. The lead heroine also has a dead father who she fondly remembers. It seems that Aliens was the influence on both movies here.
In both movies, the team is going into an Alien temple. You can argue if the Prometheus structure is a temple or a engineering base but Giger designed it as a temple. The temple has murals with alien eggs/black goo and xenomorph figures on the wall. Both movies are using an unused idea from the original Alien script that had a temple instead of a Derelict ship.
In Prometheus, 2 of the scientists get lost in the temple and stumble into the mural room. Both get infected or die. In AvP a mercenary and scientist get lost and stumble into an egg room. Both get infected and die. In AvP, getting lost in the temple actually makes sense because the temple structure is changing. In Prometheus they have 100 years better technology (3d mapping) and they still get lost.
The final bad guy survives a big explosion and comes after the lead heroine. In AvP it is the queen, in Prometheus the engineer. They have a short fight and heroine prevails thanks to some help from a "friendly" alien.
The final survivor is a woman who is left alone in a hostile environment. She is far from civilization and can only hope to reach it. In AvP it can be speculated that Lex took a snow cat and drove it to the ship. In Prometheus, Shawn took a derelict ship and flew away.
One of the bulky baddies gets facehugged during the movie. The movie ends on a shot of a chestbursting special type of Alien who screams into the camera while the inner mouth extends. The cliffhanger ending was a sequel bait. Too bad both movies had quite diminishing sequels. The possiblity of an Alien vs. Predator 3 or Prometheus 3 are quite low.
There you have it. It can be said that some of the similarities are due to the influence of Aliens (soldiers on a mission) or using an unused concept of Alien (the temple instead of derelict). The other similarities are just too big to ignore. Prometheus might be the slightly better movie, but this is due to the visual direction of Ridley Scott, no the actual story.