From Anonymous: Amour Propre and Amour Soi

One view:

Essentially, the opposite of self-preservation (amour de soi), amour propre is an acute awareness of, and regard for, oneself in relation to others.

While the savage person cares only for his survival, civilized man also cares deeply about what others think about him. This is a deeply harmful psychological deformation, linked to the development of human reason and political societies.

At its root is a difference between being and appearing. Savage man can only "be", and has no concept of pretence: civil man is forced to compare himself to others, and to lie to himself. Rousseau traces the development of amour propre back to the first village festivals, in which competition to dance and sing well increases the villagers' awareness of each other's talents and abilities. Amour propre is best expressed in a society in which wealth dominates; there, all are compared on an insubstantial and harmful basis.

A different view:

Propre can combine with a possessive adjective to mean 'my own ...'.Je vole avec mes propres ailes (I am flying with my own wings).

Rousseau distinguishes between 'amour de soi' - which we would usually translate as 'self-respect'; and 'amour propre' - nearer to being 'selfishness' or 'arrogance'.

Amour de soi tells you that you are valuable, and so are other people. It encourages you to take care of yourself, but not at the expense of doing harm to others. Amour de soi tells you that you are as good as other people.

Amour propre is selfishness.

It tells you that you are more important than other people (in most cases it tells you that what other people need or desire doesn't matter at all). Amour propre tells you that you are the most important person in the world, and that nobody else will ever matter as much as you do.

Amour de soi is not normally competitive; amour de soi encourages you to share with other people, so that both of you can benefit.

Amour propre is entirely selfish; amour propre tells you that anyone else' gain is your loss, and insists that you behave selfishly at all times, and spitefully when you think you can get away with it.