The difference between amour de soi and amour-propre—both of which mean, essentially, "self-love—is a significant one. To understand the importance of this distinction made in the Discourses on Inequality, it should be understood that Rousseau idealizes what he viewed as man's natural state, free of society. In this state, he claims, man was free to take care of himself, "his first care [was] that of self-preservation." Rousseau characterizes this as love of self, or amour de soi. He views this as a natural, healthy (indeed, essential) orientation toward the world. Men were free, only limited by the restraints placed on them by nature, and they assessed their own selves in terms of their ability to meet these needs.
Over time, though, men would change. They would begin to associate together, first as families and then as villages, cooperating on the hunt as they realized they could become more effective by doing so. In time, this led to a transformation in how they thought of themselves. Within these societies, men would begin to view themselves more in terms of how others viewed them. This might have to do with their ability to attract sexual partners, their ability to lead others, or other factors.
But the point is that this understanding of self was derived from interactions with others. This regard for self on the terms of the perceptions of others is described as amour-propre by Rousseau. He regards this development as an unhealthy on—a fundamentally corrupting consequence of living in society with others—and, in fact, the very origin of inequality itself, since it causes people to amass property in order to gain standing vis-a-vis other people.