A moot point refers to something irrelevant or insignificant.
If you're planning on buying a Ferrari with your lottery winnings but you don't win, which color car you want is a moot point: it doesn't matter. Use moot to refer to something that's not relevant to the current conversation, especially in a case where changing circumstances — like not winning the lottery — have made something moot.
Moot originates in legal language, which is why it's often used to refer to arguments or court cases. The examples below show how the expression can be used in both a legal context and in a more general sense:
Both the Redmond tech giant and the Department of Justice agree that a recent act passed by Congress makes the court case, which concerns cross-border digital privacy, a moot point. (Seattle Times)
Whether the intervention will help or hinder the pro-Europeans in Britain's Brexit debate is a moot point, as the British are prickly about anything that could be construed as outside interference in political discussions. (New York Times)
Mute means "unable to speak; silent." It doesn't make sense, then, to refer to "a mute point," unless you're pointing at something while not saying anything at all. Remember that a moot point is an argument that's irrelevant, not an argument that's silent, and you won't make errors like this one:
John Elway, the starting quarterback when the Broncos won their two previous Super Bowls, in the 1997 and 1998 seasons, thinks it is a mute point. (Reuters)
Calling something a moot point is a quick and powerful way to dismiss a line of reasoning or a hypothetical that has no bearing on the subject at hand.