The power of first person is that it makes you feel like you know the character, and are right there with him or her in the story. It's more personal and intimate. The advantage of third person is that we get the perspective in a more unbiased way.
When I teach point of view I always get my students to think in terms of a spectrum of awareness, with first person being placed on one side, the omniscient narrator being placed on the other, and 3rd person limited in the middle. The spectrum relates to what we see as a reader. Of course, in 1st person we see exactly what the narrator sees. Omniscient gives us the same god-like view that the narrator has - into every thought and character. 3rd person limited gives us more than 1st person but less than omniscient. Hope that helps!
In a first person subjective point of view, the story is narrated by the character using “I” and “me”. This narrative mode focuses on a specific character’s thoughts and allows the reader to hear the feelings and thoughts of the character. This is especially useful in a book where the character undergoes a lot of personal growth, such as Mark Twain’s “Adventures of Huckleberry Finn.” It can also allow the reader to really get to know the main character on a personal level, and can help in the characterization of a protagonist. The disadvantage is that the character might not have knowledge of everything that is going on in the story. Sometimes this can be an advantage of using first person, if the author wants their audience to discover information as the protagonist discovers it. A third person limited narrative is when the point of view is from one single character’s thoughts, and the reader is only limited to what that character thinks and feels. It is very similar to a first person narrative but the narrator uses “he” “she” rather than “I” or “me.” This point of view has the advantage of allowing the reader to find out information that the protagonist might not know.