We have 2 Legendary Super Saiyans now, Broly and Kale. Both do the same things, tho Kale does reach Legendary Super Saiyan 2.
Super Saiyan = blonde hair, power level x 50 of your current level, must maintain anger to keep the form
Legendary Super Saiyan = green hair, power level x 50? x madness, lose self control and grow based on how berserk they get.
Legendary doesn't seem to have a cap on power-like the Hulk from Marvel, it just gets stronger, but at the cost of your self. It can be overpowered, but has to be hit in 1 shot. Legendary 2 seems to grant a multiple of the current level of the legendary super saiyan X an unknown amount, and preserves the Saiyan's mind.
In the movies, Broly consistently powers up beyond his current opponent. The opponent has to draw that power out of him however. Vegito was far beyond Broly during their fight, but Broly kept getting stronger slowly catching up. Vegito had to overwhelm him to bring him to defeat. Similiarly Kale-slowly grew in power, before getting knocked out by Jiren(who was well beyond anything she had hoped to reach)
The other Saiyans, use the additional transformations to reach higher or better levels of power. Legendary is just get mad get strong