I think it really depends on what culture we are talking about. If we are focusing on the United States then I would say the biggest difference is that slavery was legal in the past and it is illegal now.
Slavery was also viewed differently in the past. It used to be acceptable and common practice. The wealthier a family was, the more slaves they owned. This was very common in the south.
Today, slavery is used in a different context. When I think of present slavery in the United States I think of children from other countries being forced to work as slaves privately in households. There have been cases recently where children have been found living in squalor being forced to clean, etc. The people who "purchased" these children justify it by saying that they are giving them a better life since they are here in the US. Sex slavery also comes to mind. I recently watched a television program about this. Women from other countries were coerced into coming to the US by being told they were going to get a college education. When they arrived, they found out that they had to pay their way to freedom by working as prostitutes.
In addition to the other answers, consider slavery in the distant past. In ancient times, slavery was more like indentured servanthood in some cultures. That meant that slaves were only "temporary" and that once they paid off their debt, or a family debt, they were freed. This is evident in the Bible where there are many instructions for slaves to obey their masters. Slavery was not the same as in later times. Slaves were more like servants.
Many of the African cultures understood slavery to be only temporary, so when they at first sold off their own slaves, prisoners from other tribes usually captured in war (usually but not always), they expected that it would only be temporary. They did not expect that the slaves would be thrown onto a slave ship and brought to another country, oceans away, never to be seen again and never to be free again.
There are various types of slavery as well, even today. In some countries in the former Soviet Union, women and men alike are used as sex slaves. This is also true in some Asian countries. Plus, in places like Darfur and other oppressed African nations, people from other tribes and other religions are being enslaved by opposing groups and outlaw groups.
To me, the major difference is that modern slavery (at least in the developed world) is both illegal and invisible.
In the past, slavery was legal and was practiced openly in rich countries such as the United States. It constituted a large part of the economy of the United States.
Today, slavery is said to exist in the shadows. People who are enslaved are generally illegal immigrants who have been brought over by their own countrymen to work as menial laborers or as prostitutes. They are not a large part of any developed economy.
The covert nature of modern slavery might be the telling difference between the past and present conceptions of slavery. In this, the previous thoughts are really accurate. The fact that human trafficking is something that nearly everyone in public would deride and criticize helps to differentiate it from its older counterpart. The fact that governments in the past were more open and willing to sanction slavery as form of generating and expressing wealth as opposed to today where governments have to pay a certain amount of "lip service" to how bad human trafficking is while failing to do enough to effectively eradicate it might be another difference.