Dr. Eggman is a weird relative of Dr. Robotnik. Despite popular belief, they are not one and the same; they are two separate entities. Dr. Eggman is the wannabe Robotnik. He claims to have built the likes of Metal Sonic and stuff, but he didn't. He is still evil, but a lot less then the more evil Dr. Robotnik. He doesn't even care about Pingas that much, although he still does. He basically is the same as Robotnik, but is worse. He is a member of P.I.N.G.A.S.
Eggman was born a little while after Robotnik. After he saw what Robotnik did in his fight against Sonic, Eggman wanted to take his place. He had You kidnap Robotnik, and took his place in a game, where Sonic called him Eggman, so he decided to make that his nickname. He has since overshadowed Robotnik, and Robitnik doesn't make appearances anymore.
He is very similar to Dr. Robotnik, but he is a lot less smart and evil. His plans are a lot more silly and wacky (in some fronts). Due to this, he is constantly looked down apon by most of the Robotnik family. But when he really wants to be evil, he can be nearly as evil as Robotnik. He blew up part of the moon (though it was fixed by Sunky), and turned Sonic into a werewolf. He is determined to make the entire land his own amusement park.