Point of view may be the most important component of fiction. So let's take a look.
First person means we are in the mind of the main character. We are going through what her or she is experiencing.
In third person point of view - we are in the mind of the narrator. To keep it simple we will say we are in the mind of an omniscient narrator. An omniscient narrator being an all seeing, all knowing narrator i.e. the narrator can be in the mind of every character in the story.
Thus in first person we are limited to a single point of view. The disadvantage of this is that we only get on side of the story. We have to trust the main character completely that everything he or she says or thinks is accurate. Well when a person is emotionally involved they are very often not as rational as they ought to be. In third person we get other ‘opinions’ on the situation. We see it from a number of sides and get a more complete picture. We may see the situation more ‘truthfully’.
However we are less attached to the story - more detached. We are not as emotionally involved any more. If we are in first person, we go through what the main character is going through. This is why ‘coming of age’ stories are invariably told through first person point of view.
Advantage: This first person, ‘eye witness’ account gives immediacy.
Disadvantage: The narrator has limited knowledge and is biased. Thus the narrator cannot interpret the events as well.
Third person omniscient gives a greater dimension to the story, making it more flexible for the writer and potentially richer for the reader. The disadvantage is the distance created by the omniscient narrator which can come between the reader and the story. Moving from character to character can also affect the unity of the story.
So it comes back to the intention of the story. Is it more important to be in the mind of the main character so that the reader can experience what he or she is going through, or do we want to get a fuller picture of the story with a little distance from the main character?
How do we know? This can be tricky. The main thing is to write a draft of the story taking a shot at point of view. Then look at it and ask the two main questions: 1) What is the story about?; 2) Whose story is it?
Answering these questions can provide the answer to point of view. But that is most probably another question.