They are different names for the same character.
The Wikipedia article you linked has the explanation, sourced from the former head of Sonic Team, Yuji Naka : "Robotnik" is the character's real name while "Eggman" is a "common name taken after his shape."
In various iterations of the franchise, they've chosen different names for him, with Robotnik being more of the English name, and Eggman the Japanese.
However, since Sonic Adventure (1999) he tends to go by Doctor Eggman in both English and Japanese translations of his video game appearances.
As far as whether or not this is a preference, a teasing nickname, or just a localization hiccup, the character rarely remarks about his different names or his own preferences.
However, as Jupotter points out , in Sonic Generations, Classic Tails refers to him as Dr. Robotnik, and he comments that nobody calls him that anymore.
This would seem to indicate that he's at least aware of his different names.
gaming.stackexchange.com/users/13845/agent86
At the risk of showing my age, geekiness, and awesome memory... Just before, or just as Sonic
The Hedgehog was being released on the Megadrive, a publication called Mean Machines (UK) had a pull out comic (Sonic The Comic didn't appear for a couple of years after that).
In this pull out comic, Sonic starts out as being brown, and cobalt in a treadmill explosion turns him blue.
Why there's cobalt involved in a treadmill, I have no idea.
In the same explosion Doctor Kintobor turns evil for some unexplained reason.
It was a really flaky story.
So, becoming his polar opposite in a flaky story, Kintobor became Robotnik (much like the Kaleds became Daleks, for all of you Whovians out there).
After typing "Kintobor" into a search engine, I have found this image:
So, based on that: his name is Robotnik.
Although, I suspect his original name was Eggman.
gaming.stackexchange.com/users/47557/Nick Tester
Same Person different name.
This name is Ivo Robotnik, he's called "Eggman" because of his shape.
Really a nickname that's been with him since Sonic adventures.
I don't remember him having a preference or not liking the name "Eggman".
He like some of the characters, does have alternate universe version of themselves.
Like "Dr. Eggman Nega".
gaming.stackexchange.com/users/24806/Ryiz
Ivo Robotnik and Eggman are actually one and the same.
Check this link for references.
gaming.stackexchange.com/users/24290/Fritz
They're actually different guys... who are the same guy:
Dr. Ivo Robotnik is a nemesis of Sonic's.
Dr. Ivo "Eggman" Robotnik is the alternate history version of the first Dr. Robotnik, who eventually crossed over to the main timeline to become a nemesis of Sonic's after the first one was finally defeated.
So, two different guys, but kind of also the same guy.
But not.
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Doctor Eggman seems to be a nickname that both he and the characters around him have accepted and use frequently.
However, Robotnik is his given surname.
In both the Japanese and American version of Sonic Adventure 2, Robotnik has a grandfather by the name of Gerald Robotnik .
gaming.stackexchange.com/users/55285/Zibbobz
It's the same idea as Princess Toadstool, vs. Princess Peach.
In the west, Peach was essentially the name she was deemed with, after the first few Mario games.
Wheras in the West after some Sonic Games, people often called Robotnik, "Eggman" based on his shape.
I think it's an east vs.
West thing.
gaming.stackexchange.com/users/142269/Janet Alex.
Are they the same, or alter egos? Does  "Eggman"  refer to a specific persona with unique characteristics compared to "Dr. Robotnik"? Does the name change based on between the 3D/2D game perspective, or the game's region (Japan vs. US)? Does he ever assert a name preference, and does he accept or reject the Eggman nickname? They look the same—is there a reason why the Sonic series uses both names simultaneously?
What is the difference between Dr. Robotnik and Eggman?
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gaming.stackexchange.com/questions/64791
gaming.stackexchange.com/users/21177/Wolf
They are different names for the same character. It seems that both names are defining the same character.
Both names refer to the same character.