Along with the underlying issue of slavery, Twain spends a great deal of time describing and portraying the attitudes that people had towards not only slavery, but the very idea that certain groups of human beings required different laws, different attitudes, and were not necessarily compatible with everyone else. If you look at the chafing between Huck and his father when Huck learns to read and to educate himself, or the conflict between the widow and Huck as she tries to understand this obviously bright boy that doesn't always want to follow the rules, you see that a difficulty in understanding each other is a common theme, pre-civil war and now.
Another aspect that is portrayed that isn't obvious is the lack of government authority that existed in much of the South and much of the US at that point. The fact that so many conflicts and problems had to be resolved locally, depending on local resources and local intuition is a part of the country's history that is gone now with our centralized schooling, centralized law-enforcement, etc., etc. Twain brings out the way that people on the frontier had to work out their own problems both with their land, their making a living, as well as with other families or groups of people around them. The constant change led to all kinds of interesting problems but also a constant state of innovation and compromise evident in the story of Huck and Jim and their adventures.