My experience has always been that Cubs fans don’t care about the White Sox but Sox fans hate the Cubs with a passion.
Yankees fans don't really care and prefer to scoff and mock at the Mets and their fans when they encounter them, while many Mets fans are obsessed at all times with hating the Yankees.
When the Mets play a game - against any team - there are always Mets fans at Citifield who start chanting "Yankees suck."
The reverse does not happen at Yankee Stadium unless the Mets are actually playing.