The main issue wasn't so much between the KMT and the CCP so much as it was between the KMT and the KMT. There were two real factions to the KMT around this time: the right-wing faction led by Chiang Kai-Shek, and the left-wing faction around Wang Jingwei. Wang Jingwei, while he despised Communism, worked closely with the Communists, and moved the government to Wuhan to compete with Chiang's powerbase in Guangzhou. He then plotted to arrest Chiang in order to consolidate his power in the KMT. Chiang found out about this, and decided to take out the Communists as he had been suspicious of them for a long time (as he felt they were Soviet puppets) and felt that they were working with Wang against him. For that matter, Wang also found out that there was a plot between the Soviets and the CCP to take him out as well. The end result was a purge of all CCP members from both factions of the KMT party. In response, pro-KMT CCP members were similarly ousted, and the CCP began organizing military forces to rebel against the KMT. This started the pre-WW2 phase of the Chinese Civil War.

You really have to understand that, from the point of view of the KMT, the CCP, which had been a small and relatively new revolutionary group, seemed to be just an arm of the Soviet (and by extension, Russian) forces that sought to influence China in their favor. The KMT was extremely suspicious of foreign interference with China (especially Chiang), so it's hardly surprising that a split occurred.

Sources:

Chiang Kai-Shek, Soviet Russia in China

Taylor, The Generalissimo: Chiang Kai-Shek and the Struggle for Modern China