Infinity gauntlet when combined with stones is a very powerful item and it gives the holder can do anything he wants. Odin is the ruler of Asgard and protector of the Nine realms. By show casing that he had fake gauntlet, he is giving hope to Asgardians and nine realms that it is with him in safe hands. If he doesn't do that, Asgardians and other realms would know that Asgard is not protected anymore. Taking advantage of this situation, enemies would wage a war. (We saw in Thor:The Dark World that Asgard was attacked.) That would create chaos. His throne and power could be in danger. To avoid all this chaos, Odin put a fake gauntlet in his vault.
In an interview to /Film, Kevin Feige, president of Marvel Studios explained this theory.
As we learn in this movie, he vanished his firstborn daughter because he sort of got what he needed, he got to the top of the Nine Realms and it was like oh, this is much too violent. And Hela says, glad to have it, ashamed of how we got it. And we see that mural in the movie of [how] it was not a pleasant history of how Asgard got all that gold. So Odin has a history of doing what he wants to do to maintain power.
“Well, it’s fake.” Because if the Asgardians knew that there was something that had that kind of power, that could theoretically wipe out Asgard and whatever else with a, in the comics, a literal snap of the fingers. They might question Odin’s ability to protect them.
So, to ensure Asgardians that infinity gauntlet is in safe hands, Odin put a fake one in his vault. Moreover, who doesn't like fancy antique pieces in their vault?