My breakdown opinion...
July
  • More Crowded
  • More locations likely available for fishing (great option if certain areas of the park are raining)
  • Warmer weather
  • Great Surface fly action with Terrestials

September
  • Less Crowded
  • Chance of Snow
  • Cheaper Hotel rates
  • Easier camping opportunities
  • Chance to fish spawning brookies in Firehole and other high water areas.
  • Chance to fish spawning Browns potentially.
  • More potential interactions with wildlife on back country rivers. (especially grizzlies)