When asked about his portrayal of the Democrats in the movie Lincoln, Director Spielberg said "you have to understand that the Democrats and Republicans have switched places since then." There are a lot of ways that this can be taken. Some of them are true and some are insulting.
In the movie, as in history, the Republicans were easily convinced to support the 13th Amendment which outlawed slavery. The Democrats were solidly against it so Lincoln and his operatives had to pick up votes one by one.
The most insulting implication is that the Democrats were on the wrong side of an important issue in the 1860s but that the Republicans are the ones on the wrong side now. You could refine that to refer just to civil rights. A reasoned argument can be made that the Republicans' position of equality before the law is morally valid and that the Democrats' position of racial preferences is divisive and harmful to the people it is trying to protect. Regardless, it gives Democrats a chance to feel smug to think that they are morally superior.
There are some ways that the two parties actually have traded positions. During the 19th century, Democrats believed in states rights taking precedence over federal law. This allowed slave states to exist. Lincoln changed that, giving the federal government primacy. He changed us from "These United States" to "The United States". Later presidents used this to expand and consolidate federal power. Since Reagan's alliance with the Libertarians, Republicans have felt that the federal government has expanded too far and are in favor of returning some powers back to the states. One reason for this is that federal one-size-fits-all programs are often a poor fit for most of the country.
There are other issues that the two parties continue their 19th century support for. The Democrats have always been the part of the poor and supported measures such as protective tariffs and market restraints. The Republicans have been in favor of open markets and free trade going back to their roots as the Whigs (the Republicans were formed from members of the defunct Whigs and abolitionists). This has continued to today with President Obama running a campaign on class warfare.
In all, it is a mixed bag. The parties have switched positions on some issues and are consistent on others. On the definings issues of the parties, they are really about the same. The Democrats are still about intervening to help specific constituents and the Republicans are in favor of free markets.