Intimidate works in every Battle, unless the opponent has Clear Body, White Smoke, or Hyper Cutter.
Rivalry only works on Pokemon of the opposite Gender.
Intimidate lowers the foe's Attack stat, and Rivalry raises your own attack.
They seem pretty equal, but Rivalry is way too situational, since there's a good chance that the Pokemon will be of the opposite gender or have no gender at all.
Intimdate is only negated by a handful of Pokemon, while any Pokemon can have the potential to counter Rivalry.