Are there fundamental differences between the adaptive immune systems of higher primates and other mammals?
I am not aware of a discriminatory feature between primates and all other mammals. However, there are significant differences for some animals. For example, the anti bodies of camelids and sharks consist of the heavy chain only, or the antibodies of sea lampreys are not of the immunoglobulin protein family.

Those differences have been exploited for biotechnology.

consider this review: http://www.nature.com/nbt/journal/v23/n10/full/nbt1127.html