Breakfast at Tiffany's is told in the first person from the point of view of an unnamed narrator. And the narrator in this story is interesting since he is telling us his story but, in the end, the novel is mostly about Holly. In this respect, he would seem to operate as a third person narrator, since in many ways he's telling us someone else's story while relating his own. But the novel is framed around the narrator's experience in New York, his experiences with Holly, and his telling of the events that occur (which is what makes this first person). There are definite advantages to this since Holly would probably be an unreliable narrator of her own life. And since so much of Holly's character depends on her interaction with other people, it seems only fitting that we get to know her through the eyes of someone who has the good luck (or misfortune depending on which camp you fall in) to know her.
There are disadvantages (or at least complications) associated with a first person narrator, though. We only directly get the unnamed narrator's view of Holly and of the events that occur, and he's clearly emotionally involved in what's going on, so we should probably question how accurate his recollections are. His views of Holly are probably very different from Fred's or Doc Golightly's or Joe Bell's, but because we have a single point of view, we don't get to know these other perceptions. And since he can't be with Holly every hour of every day, and since we know that she isn't very forthcoming with truly personal details about herself, there's a lot we probably don't get to know just because we're not in her head.
This isn't to say that Capote made a bad choice or that one type of narrator is better than another, but it does allow us to question what we gain and what we lose with a first person narrator, especially when that narrator is not the central character of the story.