Continuity mistake: The microphone on Red Leader's helmet switches from one side of his face to the other, then back again as he pulls out of the Death Star trench (just before he crashes). (Actually the entire shot is mirrored. Check out the scuff marks on the centre of his helmet, they change direction as well.) Other shots and pilots' mics switch too: Red Leader when he says, "All wings, report in", Wedge after a shot of Leia and C-3PO monitoring the battle, Luke when Red Leader says, "There's a lot of fire coming from the right side of that deflection tower" and when Luke says, "I'm hit, but not bad".
Revealing mistake: When Luke, Han, and Chewbacca attack the prison of the Death Star, in one shot, the end of the corridor is in a slope while it is supposed to be horizontal. (It looks to me like it's actually a flat panel painted to look like the corridor. The angle looks perfect for the opening shot of them entering the prison floor, it just looks "off" for any other angle.) This is fixed on DVD.
Continuity mistake: In the cantina, the alien who eventually gets his arm cut off is hassling Luke. In one shot he puts his hand on Luke's shoulder, but in the next his hand's nowhere in sight. (00:44:40)
Continuity mistake: When Luke is about to climb the ladder to get into the X-wing, there's a man fixing the ladder who starts to walk away, and another man walking under the wing and leaving. A shot later, the man by the ladder is gone, and the second guy is metres behind, though now he walks away going straight ahead. (01:43:42)
Continuity mistake: When Luke sees Red Leader hit the Death Star surface, an explosion appears out of nowhere and the X-Wing is not seen at all. Furthermore, the plummeting ship would have spread debris over a much wider area considering it was falling at such a shallow angle. (01:48:30)
Continuity mistake: When they are leaving Mos Eisley Han wears gloves, but when he pulls the lever to jump into hyperspace, they disappear. (00:54:15)
Continuity mistake: When R2D2 is captured by the Jawas, a Jawa welds a restraining bolt to the middle of a panel on R2. When the shot changes to a close up, it's being attached to the top of the panel. The shot changes again, and it's welded to the middle again.
Other mistake: When Han Solo, Luke, Obi Wan, C3PO and R2-D2 are on their way to the Millennium Falcon for the first time there is a scene where they have to walk down a few stairs. If you look carefully you can see that R2-D2 is struggling to get down the stairs but in the next shot he's down with the others.
Continuity mistake: When Luke and R2 are doing the trench run, a tie shoots at Luke and blows the top of R2's head off. But in the next scene R2's top is on again.
Continuity mistake: When Luke sees the holograph of Princess Leia for the first time, and says, "Who is she, she's beautiful," Mark Hamill's hair is rather tousled. In the next shot of him, it's neatly combed. Then, when he gets up and says, "Well I don't know anyone named Obi-Wan...," his hair is tousled again!
Continuity mistake: When Luke has been knocked unconscious by the Sand People and Obi-Wan appears and scares them off, there's a shot of Luke lying there and one of his sleeves is in disarray, halfway up his arm. But when Obi-Wan checks his pulse, he does it right through a fully-covered-by-the-sleeve wrist. (00:28:25)
Continuity mistake: In the scene just before Luke goes to eat dinner, C3PO has a towel or rag in his hand in some scenes, and it is missing in others. This continues as he admonishes R2 about replaying the message.
Continuity mistake: After Vader says "I have you now", in the first frame of the next shot, the TIE fighter disappears for the first frame and reappears in the second frame before it's blasted by Han Solo. It was restored in the first frame on the DVD.
Revealing mistake: Before Darth kills Obi-Wan, he rises his saber to prepare to die and his face and hands are perfectly visible. Right before Darth kills him, there's a wide shot where his face is missing and a black object is used to hold the saber, thus revealing the actor was replaced by an empty cloak to achieve the vanishing effect.
Continuity mistake: When the movie starts, during the shooting C3PO and R2D2 come out from a corridor on the left. R2 is looking to the right. A shot later he is looking to the left.
Continuity mistake: Darth Vader fires at Luke's X-wing and blows R2D2's top off. Half a second later it shows no damage at all. It's not even scorched. When Luke lands, R2D2's head is destroyed and it's scorched all over. (01:56:15)
Answer: It is heavily implied that one who becomes a Force-Ghost achieves an untold level of power upon entering the state. Given that they have become a pure entity of the Force, it seems to back up the statement. Obi-Wan also becomes free to assist Luke in any case.
Darius Angel
In addition to this answer, I think Obi-wan also became powerful because he "let go." He did not feel a need to defeat Vader on this occasion - he was willing to surrender the fight and "move on" to a new state. Something that Vader might not currently understand.