What is the difference between Dr. Robotnik and Eggman?

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.

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.

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".