I don't believe that there is a "one size fits all" answer to the OP.
HS have rivalries. There are some rivalries in club, but not to the extent that there are in HS. Definitely more emotion plays in to the equation with those HS matches.
There is also the "Senior" complex with HS. They want to go out on top, etc. Obviously, not the case with club.
But, there are club matches that have plenty of emotion: state cup, tourneys, regionals, nationals, etc. I think it depends on the emphasis the coach puts on the matches.
