Unlike how some discs are mastered where the alternate beginning/ending is an alternative choice when watching the entire feature, the Prometheus Blu-ray includes the alternate beginning and the alternate ending in the Extras sub-menu as individual clips. So I didn't have to worry about which way to watch the movie: the theatrical way or the alternate way. The only way to watch the entire film is the theatrical release.
The alternate beginning is about 2 minutes in length, and the alternate ending is about 5 minutes long, so they're not very long at all.
The alternate beginning and ending scenes were sanctioned by Ridley Scott. The alternate beginning was a slightly expanded version of the scene with the engineer in the beginning. The alternate ending was an expanded version of Shaw and David's conversation prior to Shaw leaving. It was based on Ridley Scott's original title for the film.
Therefore, these scenes are canon. Furthermore, having watched them, they don't change anything about the plot. They just have different visuals and different dialoge.
For more information about the beginning, see the following spoiler:
The menu before the alternate beginning reveals that the Engineers are seeding Earth with life, something which was questioned by many viewers. The actual text of the menu is:
Scene 7: Arrival of the Engineers
The Engineers touch down on prehistoric Earth for a sacred ritual devoted to the seeding of Intelligent life. Whereas the Theatrical version of the scene is devoted to the Lone Sacrifice Engineer, this early cut features several of his comrades, including the Elder Engineer. The ceremony was filmed with dialogue but unsatisfied with the result, Ridley Scott removed the dialogue entirely.
This sounded to me like the scene would contain dialogue. However, apparently Ridley Scott was so unhappy with how the dialogue turned out that he removed it altogether. All the beginning shows is a few more Engineers on Earth along with the Lone Engineer, and different shots of the Lone Engineer's sacrifice.
And for more information about the ending, see the following spoiler:
The description of the alternate ending from the Blu-Ray's menu reads:
Scenes 141-150: Paradise
This extended scene reveals new information about the Engineer Homeworld as well as some alternate dialogue between Shaw and David. Note that one of Ridley Scott's original titles for the film was Paradise and it was even maintained as a cover title for secrecy during production.
In the scenes, we see an extended version of Shaw uprighting the vehicle she uses to drive to rescue David with more dialogue where Shaw demands to know what the Engineer said before he ripped David's head off. (This exchange was shown in another deleted scene) Then, upon arriving at the ship, Shaw asks David if he can use the Engineers' maps to navigate to where they came from and he says he believes he can (just like before) and goes on to ask why she wants go to. He reveals that there is no word in Human language to describe where the Engineers come from, but the closest would be the word "Paradise"
So it's the same plot (Shaw and David leave for the Engineer Homeworld) just new information about what they might find there.