So if I play Nico Bolas the Planeswalker, does he not die if a Nico Bolas (the creature) enters play?
The Legend rule cares about card name. One is named "Nicol Bolas," the other is named "Nicol Bolas, Planeswalker." As much as we all know they're the same dragon, for the purposes of the game, the 2 have as much in common as "Forest" and "Progenitus."
The "planeswalker rule" cares about planeswalker subtypes. "Nicol Boas, Planeswalker" has the planeswalker subtype "Bolas," and "Nicol Bolas" no planeswalker subtype at all. Ajani Goldman and Ajani Vengeant both have the planeswalker subtype "Ajani,"so if both are on the battlefield and the same time, they both put in the graveyard.