Elections always have consequences. While some may think both major parties essentially the same, the fact is that the ideologies are fundamentally different and will carry different influence over the functions of government. Also, not all government positions are elected positions, there are many important roles that are filled by appointees which are confirmed by congress. While some roles are non-partisan, many are partisan and the winning party will fill seats with people who have similar ideologies. This can have major impact on laws, regulations, and enforcement, as well as influence the way people feel about the government and regulations or laws.
Despite the political hype, the Republican and Democratic parties actually have a lot in common, especially when compared to other country's political differences. Thus it is easy for many Americans to believe there is little difference which party wins an election. This is not true however, as it can have a major impact. Historically, if the Democrats had won the 1860 election there may have never been a Civil War or Emancipation. If the Democrats had won in 2000, there probably would have never been an Iraq invasion. So elections are important.
Whatever party wins is the one whose ideas will rule. Ideas have consequences. When the Nazi's won the elections in Germany, a much different outcome was created than if a less jewish hating organization had come to power. The particulars of a particular party are the determining factor. To remain ignorant of that is what led to WW2.
I think overall, it does make a difference when a particular party wins. When democrats win, a lot more social issues and bills are passed to pander to their voters. I don't think it's that major when either party gets elected, but there is a benefit to having your favorable party winning.
Not long ago you could see the differences between political parties, and you knew pretty much what to expect from the winning party. So much has changed. It does not matter anymore which party wins. Politics has become an arena where the 2 major parties start and end their days in battle, with each party trying to make the other look inept - and we the people suffer for it.