Looking over both boards, it seems that BYU fans poke fun of the state of wyoming, and poke fun at things such as higiene and education. Wyoming fans ridicule BYU's religeon, and are generally obsessed with hating BYU. (in sort of a weird angry stocker sort of way. Where BYU fans are 90% unaware that Wyoming even claims some sort of rivalry with BYU).
Byu fans make fun of wyoming mostly because while wyoming hates BYU, BYU generally does not care about wyoming very much. Though many BYU fans have some very bad experiences with UW being violent or over-the-top unsportsmanlike when attempting to travel to wyoming for a game.
When BYU makes fun of UW, its mostly harmless fun.. When I see Wyoming talking about BYU, it does come off the same.