What is the difference between a society with slaves and a slave society?
The distinction between a "society with slaves" and a "slave society" was first made by scholars of the ancient world. When it was adapted to the context of the Atlantic World, historians like Ira Berlin have used it to describe the difference between societies like colonial New York, where slavery was widespread but not central to the economy or to social structures, and the Carolina Low Country and the Caribbean, which were entirely built around the institution. At least in the colonial period, slavery and unfree labor were important almost everywhere. But in "slave societies," the laws, the economy, and even politics were structured around the institution of slavery.
Usually the establishment of a slave society occurred in response to economic forces. For example, in the years following the founding of Jamestown, the Chesapeake emerged as a slave society due to the importance of tobacco as a cash crop. Similarly, the profitability of cultivating rice in the Low Country, and sugar in the Caribbean, led to the establishment of enormous populations of enslaved people there.
Percentages are one way to think about the difference between a "slave society" and a society with slaves," but this model can be problematic in the British American colonies, because slave populations were far more dense in the Tidewater and coastal regions (where cash crops were cultivated) than in the Piedmont and mountains. By the outbreak of the American Revolution, over 50% of the population of South Carolina was enslaved, but this number does not accurately reflect realities. In the upcountry, there were relatively fewer slaves so we could describe this as a "society with slaves." Some counties in the Low Country, however, were more than 98% enslaved. In the Atlantic World (unlike antiquity), the fact that slavery was tied to race meant that a "slave society" had, as historian Peter Wood described it in a book by that name, a "black majority."
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.
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.