The difference between these two types of society has to do with how central slavery is in each.
In a society with slaves, slaves are an incidental part of the society. The society's laws and its social structure are not built around slavery. There is slavery, but there are other kinds of labor as well. By contrast, a slave society is completely built around slavery. Slavery is integral to the way people think and the way they act. This is a society that could not exist, economically or socially, without slavery.
Perhaps it will help to think about an analogy. This is the same as the difference between a person who goes to church pretty much every week and someone whose life is completely devoted to religion. In one, religion is a part of life; in the other, religion is the basis of life. The same applies with the two societies in your question.
This is a good nuance to make. All socieites have a "slave" population. It might be surprising, but human trafficking is something that is more common than might be expected even in the United States. Recently, this horrible blight of human civilization is getting more attention.
From an academic point of view, scholars seek to answer this question by percentages. In other words, if a society has 20% of it population as slaves, then they deem it a slavey society. Of course, not all scholars will agree with this number, but there is a decent amount of consensus. More important, there is a theoretical difference between slave societies and societies with slaves.
Based on this criterion, there have only been five slave societies: ancient Greece, ancient Italy, United States, Brazil, and the Caribbean Islands.
I will also add a link to Orlando Patterson's excellent book on this topic.