Clásico is just the Spanish word for Derby. It usually means the same city (Rangers-Celtic in Scotland, River-Boca in Argentina, Millonarios-Sta. Fe in Colombia), but not always (Southampton-Portsmouth as the "South Coast Derby" in England, Bucaramanga-Cúcuta as the "Clásico del Oriente" in Colombia).
Derby is a match between two local teams. OM - PSG or Barca - Real are Classico but both are not between locals. Maybe in spanish classico means derby, but in french and english it's just a rivalry and I think OP meant that.
I personally feel our biggest "rival" is Rochester. Games with Montreal always tend to be more heated than others too. We had a Southern Derby with Charleston but I never disliked them - I pulled for them most of the time they weren't playing Carolina actually.