Medically, there's probably no difference.

All the differences come down to the religious aspect of the Briss. The briss welcomes young jewish infants into the religion and is one of the two major events that happens in a young jewish man's life that MAKES him a Jew (the other being his Bar Mitzvah)

Edit to Add: Interestingly enough, the Briss is SO IMPORTANT that it's one of a very small list of activities that can be done during Jewish Sabbath.