Revealing mistake: When Gandalf shakes Bilbo's hand it should be a lot bigger than his but it is the same. (02:25:26)
Revealing mistake: During the battle of Dale, Bard confronts Alfrid, who is dressed as a woman and is trying to escape with looted gold. During the confrontation a troll climbs over the parapet and is killed by Bard. As the troll falls, Alfrid also falls. When he lands, the ground near his hand ripples like fabric covered in sand.
Answer: When Aragorn was 25 he left Rivendell to travel the world, having had his proposal to marry Arwen rejected by Elrond. Hence he is absent during the time period of the Hobbit, having gone to Rohan and then Gondor in disguise. During his travels he becomes very well known among the Elves, owing to Elrond raising him and Galadriel welcoming him to Lothlorien after he comes back from his long travels. Even before, he had served in the Rangers of the North as he was growing up, where he gained the nickname Strider. Legolas already knows Aragorn before LoTR, as Aragorn had initially left Gollum in the care of the wood-Elves before someone helped Gollum escape.