What are the various advantages of writing in first and third person?

Beginning Writer's Answer Bookedited by Jane Friedman

First-person viewpoint involves only one character (presumably your protagonist) and permits close reader identification. The disadvantage is that all other characters must be developed strictly through their actions and dialogue as observed by the protagonist, which can be difficult and limiting.

Third-person omniscient viewpoint allows you to get inside many characters’
heads and explore multiple viewpoints. You can also tell or show
things that would not be known to any of the characters. In the case of third-person limited viewpoint, you can show the way your protagonist perceives and reacts to events throughout an entire novel, and make him very well rounded and demanding of reader sympathy.

To choose the best point of view for your story, you might try writing several different passages, each with a different point of view and central character. Play around and see if anything fits. Don’t be discouraged if this exercise makes things worse; many great novelists come to realize the “right” point of view only after putting many months of work into a manuscript.