the Hoepli dictionary gives: councilman, city/town councillor, member of a local/city council for consigliere comunale.
For assessore comunale it gives city councillor.
Not much of a difference there, is there?

Both are elected Town Councillors. The Assessore, however, is in charge of a specific area of council administration, Sport in this case. In the UK, we tend to have employed Directors who carry out most of these functions, although they answer to a committee of councillors, headed by a Chairman/Chairwoman.
One example here:
"Chair of Recreation and Playing Fields" (precise titles vary from council to council).
http://www.saffronwalden.gov.uk/downloads/BUMBER BOOK JULY 0...