Désespérance is rarely used.

• TLF understands that it seems more pessimistic than désespoir.

• If, for Batchelor (in Using French Synonyms) désespoir = despair, désespérance = despair + lack of faith

The difference is essentially the same than between espoir and espérance.

Désespoir is common and can be used in all cases while désespérance is rare and cannot always replace désespoir.

See also http://alafortunedumot.blogs.lavoixdunord.fr/archive/2009/05/10/o-rage-o-desesperance.html

Désespérance to me has religious overtones being the contrary of espérance, one of the three theologal virtues with foi (faith) and charité (charity). Thus désespérance is the state that Christians find themselves in when they have lost hope in God.

Apart from this specialized meaning, it means much the same thing as désespoir, with the following nuance however that can be inferred from Walther von Wartburg’s remark about désespérance in his Französisches Etymologisches Wörterbuch :

• Désespérance, ungebräuchlich im 17. und 18. jh., wurde von der Ecole romantique und den Symbolistes wieder aufgegriffen, ist heute aber vor allem literarisch und wirkt abstrakter und negativer als deséspoir.

My translation :

• Désespérance, rarely used in the 17th and 18th centuries, was revived by the Ecole romantique and the Symbolistes, it is now mostly literary and comes across as more abstract and negative than désespoir.