In the sense that Vader considers himself a different person from Anakin (one would say "different, yet the same" with emphasis on "different"), does Sidious feel the same about himself in regard to Palpatine, only putting up with being called Palpatine for practical reasons?
This is explicitly answered in the official novelisation for Revenge of the Sith. In short, Sidious considers Palpatine to be his disguise (to the world outside) and Sidious to be his real self.
As @hypnosifl has pointed out in a comment, there's a very relevant quote from Palpatine actor Ian McDiarmid that states that Lucas had very much the same idea in mind; Sidious was always the true form and that he wore Palpatine's unblemished face like a mask. When he became hideously scarred, that was, ironically, his true face.
Lets consider in what way is Anakin "different" from Vader. While Anakin has some dark attitude he is mostly a light side guy. He is tragically seduced by the dark side and loses all his natural senses and good feelings, murdering children and becoming an emotionless Dark Lord.
At this point, quoting Obi-Wan:
In his redemption, on Return of The Jedi, Vader gets in touch with his deep feelings, starts to care about his son, at that moment he becomes Anakin again, a good man.
Now, when it comes to Palpatine/Sidious:
Sidious is an evil character, he cares about no one but himself. Also he is a master planner, it is all about his plan to become the ultimate ruler of the Universe.
When he is acting as Palpatine, even though he is hiding his dark nature, he is moving forward with his plan and manipulating things to work in his advantage. He is carefully inciting Anakin with the Dark Side, and creating a Galactic war to stay in power.
So, with Anakin/Vader you can see a clear distinction between the two personalities, and there's no point where they coincide.
With Palpatine/Sidious both personalities have the same goal and motivations, and he is acting as both at the same time.
So no, Palpatine is not a different person than Sidious in the same way as Darth Vader is.