http://nation.time.com/2012/02/17/japan-wages-war-over-what-to-name-the-last-one/ This article explains it at a basic level.

To give a bit more detail: at the time of the war, the right-wing, fascistic Japanese government justified their military actions in Asia by saying they were liberating it from the imperialist west. True in the sense that yes, they were expelling western imperialists, but untrue in the sense that they were fulfilling their own imperialist aims and the countries they "liberated" often hated them more than their European colonizers, and the violence committed by Japanese troops was often horrific. They had a whole imperialist ideology, according to which the Japanese were fulfilling their divine destiny to rule over an Asia united under the Japanese. They called this global entity they were creating the "Greater East Asian Co-Prosperity Sphere" (I'll abbreviate GEACPS). No one outside of Japan believed this specious logic. So during the war, the official name for the war was the greater East Asian War. This was in line with the GEACPS entity--they were waging a war to create the GEACPS. After the war, the U.S. occupiers suppressed this previous ideology and its imperialist buzzwords. So basically the Greater East Asian War was a no-no because it was a propagandistic term that supported a particular ideology.

After the U.S. left, in the ensuing decades, conservatives have been fighting ever since to propagate their own view of history, which generally supports the logic used during the war. As such, they want to use terms used during the war. Others want to use a different term, to demonstrate their disagreement with that ideology and avoid being associated with the right-wingers.

As a separate issue, calling it "World War II" is problematic because WWII is typically dated as beginning September 1, 1939, when Germany invaded Poland. But hostilities in Asia are much older. Many date the beginning of the Pacific War (or whatever you want to call it) to September 18, 1931. you could go back even further, to the invasion of Korea by Japan in 1904. Right wing people in Japan often choose December 8, 1941 as the beginning of the war, which highlights Western hostility centered around the U.S. Oil embargo of Japan, necessitating and justifying, according to this logic, the Japanese to declare war against the U.S. And sweeping under the rug, not counting as war, all of the Japanese violence in China and Korea that was the provocation for the U.S. to declare the embargo.