Back when I was a missionary in Brazil, we met this woman and her friends who, upon our arrival, asked several questions about what they thought our beliefs were. After disproving many of their ideas, like cult sacrifices and baptizing in our own blood, she then asked, "Then what is so different about your religion and mine?" We were more than willing to answer that.
Our explanation for her was that what set our religion apart from the other religions was that we had a living prophet that speaks directly to God.
But, what does that mean? It means that we believe (as Latter-day Saints) that we have a prophet, like Moses, that speaks directly to God in our behalf and receives revelation for the entire world and is the only man with the authority to do so. This is a hefty claim.
Then, why is it important to have a prophet? This is a common argument, because we have the Bible and many people believe that God revealed all that is and was necessary within its contents. Although the Bible does contain many great and important things, I think it has something to do with jurisdiction, or stewardship. The prophets and apostles were speaking to people in their day, because that's what they were called to do and who to teach to. They were revealed things that applied to them in their day specifically for their people. Yes, many of those same teachings and revelations apply to us in our day, but many of their problems were different than ours. They didn't have to worry on what to do with pornography, gay marriage, or the immense spread of technology.
I can't think of a time that needs a prophet more than now. As the times change, we need further understanding of what God needs and wants us to do. Our problems now will be different ten years from now and even a hundred years from now. As they were needed hundreds of years ago, prophets are still needed now.