Cheap labor and high profits is the main reason that different types of slavery still exist in the 21st century. It is the same reason that Americans imported slaves during the previous centuries, and it is likely that the practice will never be completely eliminated.
People have long been willing to exploit other people for their own economic advantage. As has already been implied, slavery of various kinds still occurs even in countries that have officially outlawed slavery. If it were simply a matter of waiting for "enlightenment" values to penetrate into every culture on earth, there might be some hope that slavery could finally be abolished. However, the fact that slavery continues even in supposedly "enlightened" countries is a cause for pessimism.
From what little I've read about current slavery, it seems to be a function of unchecked power and greed over necessity or blindness. We in the U.S. don't think of Illegals working in fields or brothels as slaves, but they may be giving all their earned money to their handler (read: pimp) and working in extremely poor conditions. It's not classical slavery as we understand it from our textbooks, but the base factor -- people as product instead of individuals -- remains.
In other countries, slavery continues because no one has stepped up to stop it. It took a civil war just to start the road to abolition in the U.S.; elsewhere the governments never even considered an equal-rights-based system, and so the movements have trouble generating enough steam to continue.
Because we people are power hungry and greedy. We often do not care about what happens to other people so long as we get what we want. If we go to a restaurant and we get good food for a decent price, we don't care if the people washing the dishes have been brought over by a smuggler and are being forced to work. This sort of attitude makes slavery continue to be possible.