In general, Rare Titans function the same as regular Titans. You will receive a score and rewards based on the damage you inflict. All rewards are received once the Titan has been defeated - or if undefeated - once it has escaped.
Unlike regular Titans, Rare Titans are much more difficult to defeat and appear only rarely. Compared to regular Titans, Rare Titans have much higher health, a stronger Special Skill and will reflect the damage from a certain element. Rare Titans will also offer a bonus chance to receive a high-level Ascension Item as an additional reward. Please note - this additional reward is not guaranteed as the odds of receiving the Bonus Reward is random.
Rare Titans appear only by defeating regular Titans.
If you have joined an Alliance after the Rare Titan has spawned, you will receive reduced titan loot and will not be eligible for the bonus loot. This includes the chance to receive the High-Level Ascension Item.