The strict notion of regionalism that helped to define the sides which led to the Civil War is not as present in the contemporary setting. The knowledge that slavery was the primary issue which divided the North and the South is still there, and most view that as the cause of the war. However, the strict sense of commitment to one's ideals and principles, as well as the driving force behind one's own region is not as present in the contemporary view of the Civil War. Part of this is because the nation is no longer seen in terms of regionalism as cosmopolitanism and mobility have become a critical part of contemporary American identity.
I participated in the AP US History reading several years ago in San Antonio, Texas, with several hundred other readers from around the country. The week started with several hundred of us teachers from all over the country meeting in an auditorium at Trinity University. African-Americans were sitting mostly together in the balcony, where I was at.
As the speaker introduced the essay question for the test on the Civil War, an older caucasian gentleman sitting in the first few rows raised his hand and interrupted the speaker and said with a thick southern drawl, "Down in Georgia we prefer to call that conflict the War of Northern Aggression". There were laughs and some clapping. Then an African-American man behind me stood up and said loud enough to hear down below, "Or as we like to call it, the War of Liberation". More claps and some cheers. I thought I had landed smack in the middle of a new Civil War.
My point being that each region, and population have their own contemporary interpretation of the Civil War, which is continually being taught and passed down the generations. If you looked in US History books for parts of the South, you will notice the Civil War is treated differently in textbooks, as argued above, like it was a crusade that was brutally oppressed. This isn't true so much in the larger cities, but in the smaller towns and rural schools. In the North, I have found students mostly ambivalent about the topic. They can't identify with it the way southerners still can. If you want a more detailed picture of what it's like in parts of the South still with regards to this conflict, read Tony Horowitz' book, Confederates in the Attic.
I would say that the differences in how the war is seen in the two regions of the US is diminishing (compared to how strong it was a few decades ago) but is still important.
The differences are diminishing now for two reasons. First, civil rights are not such an issue between North and South anymore so there is much less reason for the South to feel hostile to the North. Second, as the South has boomed, many northerners have moved there, diluting the culture.
However, there is still a significant difference. The Civil War is seen by many Southerners as a noble struggle in a lost cause. They see the South's cause as one of states' rights, not slavery. And they believe that the Confederacy should be celebrated. See, for example, the link below.