The two most common narrative points of view are first person and third person, and I hear plenty of fiction writers, especially new-to-the-crafters, fuss over which to use. Often, it winds up as a question of preference and polling friends.
With first person, the idea that it is immediate, close, or immersed often comes up. I don't see that. I think when third person is close and the author understands free indirect discourse, third person can be just as immersive and close. The character's voice can come through just as well.
I do find a couple of other advantages though.
Among them is that use of what I call the memoire effect. In past tense, first person can look back with a sense of maturity and experience in a way that can help a story. The counter-part to this in third person is the external narrator, who can look at a story objectively or at a distance, or can guide us on a journey through time and space to other POVs when needed. We tend to avoid it these days, but an external narrator can even act as a confidante. "What Todd didn't know was … " (And the lack of an intruding external host can be why first person feels more intimate or immediate to some readers.)
The memoire effect can be seen in Nabokov's Lolita when he says, "Lolita began with Annabel." This comes early on in the book when he is trying to explain what the narrator believes contributed to his sickness. It's a mature and introspective analysis that the young could not have offered.
We can also see this in Ketchum's The Girl Next Door when the neighbor's mother is introduced as a vile and manipulative person, but then on-scene is a beloved friend, creating a contrast between now and then. Later, the first-person narrator explores the us-vs-them psychology of the events and the small details that alienated the victim in the story as it progressed.
They're small moments that add a lot to the story.
And in both first and third person, these particular advantages (they aren't the only ones) deal with getting around the here-and-now viewpoint in some way, whether it's just switching to another character or providing a different perspective on the incidents.
But not all first-person narratives have so much chronological distance. Nabokov later switches to quoting a journal, where he only has a day's worth of perspective to add to the events, rather than years. Here, he takes on the more introspective analytical tone on the events, but without the experience of murder and rape to soften the demeanor. In contrast, it's a wonderful tour of the psychology of the character.
And some narratives skip this memoire effect altogether. A popular candidate is the Hunger Games, which led a stampede of other first-person present-tense narratives. Here, the narration lacks a looking-back introspection, though it can still be introspective. Being first-person, it skips the external narrator (notice that the movie version benefited from switching POV to give us some omniscient events.) Personally, I think this narrative mode worked for the first book, but the second and third books could have benefited from either an external narrator or a memoire effect to help cover some of the incidents.
First-person has other strengths to consider, such as the omnipresent voice and perhaps the intimacy of a one-on-one conversation. In fact, I think all narrative modes have advantages that need to be explored and understood. Too often, they're overlooked in favor of personal preference, or they're labeled with vague terms like "immediacy" that could be applied to other modes as well.
Writer of speculative fiction. Current work in progress is an untitled dark fantasy novella.