It is difficult to untangle the rhetoric from the reality when it comes to analyzing the various work-patterns and arrangements in nineteenth century America, where the debate referenced in this question was most intense. It is also difficult to generalize about actual work conditions, which varied by occupation, by status, and according to other criteria, including the personality of the slave or business owner, local laws and customs, and other factors.
Slaves were by legal definition chattel, meaning they were akin to livestock. This meant they could be bought and sold, and subjected to physical punishment, including, in some states, death. While masters could not arbitrarily murder slaves by law, most restrictions on their treatment of their slaves took the form of customs and even negotiations rather than legal obligation. Slaves did not have legal possession over their own bodies or those of their children. Families could be, and often were, split up, and the institution of marriage had no legal sanction. Masters were expected, however, to supply food, a meager amount of medical care, and shelter to their slaves, and this requirement extended into old age. The fact remains, however, that among plantation slaves, the slave community itself bore the brunt of this expectation.
The legal status of industrial workers, in theory at least, is that they were free agents exchanging their labor for compensation on a free market. While this was obviously not the case for many, particularly child laborers, workers in northern mills were not chattels. On the other hand, owners were not expected to provide care for workers, especially after they reached the age where their productivity had dropped. Many owners did exercise tight control over their workers through mill villages like Lowell, Massachusetts, and many also kept their workers obligated to them through debt in company stores. And it is also the case that the conditions in many northern mills were abysmal. The chief difference between these two exploitative systems of labor was that slaves were chattels, and mill workers were not. This was a very significant difference, as it severely circumscribed the possibilities available to slaves relative to those available to mill workers.