By telling the story from the third-person limited point of view, Jacobs is able to allow the characters to keep their internal thoughts to themselves. In doing so, he allows the reader to create them on their own.
Rather than describe the thoughts that Mr. White goes through as he considers the tale told by the Sergeant and whether or not he ought to wish using the monkey's paw, Jacobs allows the reader to reconstruct the thought process in their own heads, arguably far more effective than simply outlining it for the reader.
By recounting the conversations without the thoughts behind them, Jacobs far more effectively links the reader to the themes of greed and the inability to follow good advice sincerely given. It is easy for the reader to imagine Mr. White wishing for the money that he eventually gets but only after the death of his only son. It is easy to connect the horror of the mangled boy's body being re-animated and returning to the house and the desperate love of the mother that pushes her to disregard caution and good sense in wishing him back to life.
By using the third person limited voice, Jacobs crafts a far more effective story with these themes.