The ultimate weapons are using the original formulas, with a few differences.
Cloud/Ultima Weapon: Same as before, power decreases as his HP gets lower (the sword will slowly turn from white to blue, as a visual indicator of the amount of power lost). Note that it's the % of your HP remaining, not the actual amount; a Cloud with 300/300HP would deal more damage with it than a Cloud with 5000/9999HP.
Barret/Missing Score: Power is based on the total AP of all the Materia equipped to the weapon; Mastered Materia like Quadra, Mime, etc. tend to give the best bonuses. Does very little damage with no Materia/low AP.
Tifa/Premium Heart: Changed. Basically the same as Ultima Weapon, but uses MP instead. The more MP you use, the less power it has.
Aeris/Princess Guard: Changed. This heals targets, and grants the Shield status.
Red/Limited Moon: Changed. This deals more damage based on how high your Limit gauge is. Best used with a filled Limit Lv.4 and commands like Deathblow or Mug (as attack is disabled when Limit is available).
Yuffie/Conformer: Deals more damage based on target's Level; also ignores the penalty on Morph attacks.
Cait Sith/HP Shout: Changed. Deals damage equal to the difference between maxHP and currentHP (so an
? effect).
Vincent/Death Penalty: Changed. This now has the Yoshiyuki effect, and will gain a damage boost whenever another party member is KO'd (if both are KO'd, then damage is increased again).
Cid/Venus Gospel: Changed. Has no special damage formula, but drains damage as HP.
They also carry some stat bonuses, usually in-between the bonuses given by the other end-game weapons.