What are the differences and advantages/disadvantages of writing a characters point of view in the first person compared to writing them in the third person?
You can't have an omniscient third-person "unreliable narrator."
It is sometimes desirable to use the device of an "unreliable narrator." Shirley Jackson's We Have Always Lived in the Castle would have been a very different story without Merricat as a narrator; her eccentric worldview and selective release of information provide for a more entertaining story than a straightforward narrative of events would have. Chief Bromden, the narrator of One Flew Over the Cuckoo's Nest, is likewise mentally ill and this enhances the storytelling possibilities of the novel (although the film employed a more straightforward narrative technique).
Mark Twain's two best novels, The Adventures of Tom Sawyer and The Adventures of Huckleberry Finn , use different approaches. Tom Sawyer uses third-person narrative; his actions are a bit more interesting than his thought processes, and the story occasionally requires scenes that Tom Sawyer isn't in. Huckleberry Finn , by comparison, is told entirely from Huck's point of view and his skewed (but completely honest) viewpoint is what makes the story compelling.
Some characters are simply more interesting for their actions than their thought processes (I doubt the inner workings of Beowulf's mind would be all that revealing), and for these, a third-person narration is recommended. For other characters, it's the other way around. The sequence of events in Catcher in the Rye is far less remarkable than Holden Caulfield's description of them, and for these kinds of stories, a first-person narrative approach is preferred.