While it is true that the war was not specifically fought over slavery, I think that it is going too far to say that slavery was “just a small part” of what caused the war. Slavery was at the heart of the causes of the war.
The major cause of the war was that the North and the South did not trust one another. Each thought the other side was working to destroy the other’s way of life. While there were many aspects to these different ways of life, the most important difference was slavery.
The North and the South had very different social and economic systems. The South was based on a plantation economy that raised staple crops. Of course, the work force for this economy was enslaved. The North had a mixed economy based on free labor. The South was an aristocratic society headed by plantation owners. The North was much more egalitarian. In these ways, the two were sort of like separate countries that were officially united.
Because the two “countries” were so different, they were each suspicious of the other. They each tended to think that the other was gaining the upper hand and would destroy their way of life. The war was fought mainly because each side wanted to preserve its way of life, but the differences in their ways of live were largely centered on slavery.