From ICD-10 (International Statistical Classification of Diseases and Related Health Problems),
Asperger's syndrome (F8.45) is:
A disorder of uncertain nosological validity, characterized by the same type of qualitative abnormalities of reciprocal social interaction that typify autism, together with a restricted, stereotyped, repetitive repertoire of interests and activities. It differs from autism primarily in the fact that there is no general delay or retardation in language or in cognitive development. This disorder is often associated with marked clumsiness. There is a strong tendency for the abnormalities to persist into adolescence and adult life. Psychotic episodes occasionally occur in early adult life.
Actually, the Wikipedia article that you are pointing to also makes it clear:
It differs from other autism spectrum disorders by its relative preservation of linguistic and cognitive development.
However, there is no consensus if there is difference between Asperger's Syndrome and High-Functioning Autism. Perhaps also there is no clear distinction between clinically-significant autistic traits and complete absence of them.
Currently autism and Asperger's Syndrom alike are not fully understood on neurobiological basis (there are ongoing research, for a review see e.g. E. Pisula, The autistic mind in the light of neuropsychological studies (2010) or a special issue of Nature on autism).
However, the premise
The only criteria seems to be that people with AS are anti-social and do not have some other more serious psychological disease.
is clearly false.