Why does Darth Vader operate openly with his Sith name under the Empire, while Darth Sidious keeps his a secret in favor of Emperor Palpatine? Is there an explanation in either Canon or Legends?

Emperor Palpatine is his real name, and its who he was when he achieved power. Vader was an unknown. Nobody knows anakin Skywalker is Darth Vader. Introducing him as Vader adds mystery and menace. Changing palpatines name would cause issues of trust among the subjects Edited July 14, 2016 by BlazeTomahawk

The book Lords of the Sith by Paul S. Kemp describes how Darth Sidious undertakes thorough, broad, and excessive steps to conceal his ability to tap into the Force from the public. Darth Vader is allowed to openly brandish his lightsaber and be seen by the public and military ranks, whenever necessary, defying physical science with Force powers in service to the Emperor. Were Palpatine to start using his Sith moniker outside his utterly loyal and entirely devoted circle of closely knit confidents, which so far we know consists of some members of the red-robed royal guard and Yupe Tashu, it wouldn't be long before people worked it out and started asking questions about how much he and Vader have in common.

People didn't know Palpatine was a sith lord, the whole galaxy thought he was just an old man. The book, lords of the sith, does a good job explaining this. Only Vader and the royal guards knew that he was a sith.