Princess Sailor Moon was a form of Sailor Moon featured only in Pretty Guardian Sailor Moon. Princess Serenity's cold, vengeful, and determined personality controlled this form, rather than Usagi.
There was not much development or explanation of the Princess Sailor Moon character in the series; all that was known was that she was dedicated entirely to the cause of getting Prince Endymion back, not caring who or what was destroyed in the process. It is entirely possible that there was more to her than that, but her intense drive to reach her goal was her dominant personality feature at the time when she appeared in the series.
Unlike in the anime, the one who used the Silver Crystal at the end of the war between the Silver Millennium and the Dark Kingdom was Princess Serenity, instead of Queen Serenity. She effectively destroyed the world in her grief for the death of Endymion.
In the present, after Usagi awakened as the Princess, the Princess soon started taking control. When Usagi's emotions of sadness and loneliness got the better of her, the Princess's powers leaked out, taking control of her mind and transforming her physical appearance. As Princess Sailor Moon, she was more powerful than Sailor Moon alone, and she was ruthless in her mission to reclaim Prince Endymion and unrestrained in the use of her powers to achieve her objectives; she even injured the other Senshi and Naru Osaka in one instance. She was willing to destroy the world again if it meant being with Endymion in the next life. This was a constant battle between the Senshi and Usagi trying to convince her not to go to such measures.
Fans jokingly refer to the princess as one of the two "Ideon" characters of Sailor Moon canon (after the titular Super Robot of Yoshiyuki Tomino's series, Space Runaway Ideon), the second one being Hotaru Tomoe aka Sailor Saturn due to her destructive powers, blatant disregard of whom she hurt as a result of these powers, and, ultimately, her ability to destroy the world if and when everything came to a head.
Princess Sailor Moon wielded a very powerful sword that could produce bursts of magical energy and explosions, as well as parry magical attacks sent against her. The sword could also transform into a harp on which she played a haunting melody; it was a force of healing as much as the sword was a force of destruction.
She also had access to the powers of the Silver Crystal which was inside Usagi's body.
Princess Sailor Moon also appeared to be capable of telekinesis. In the final episode, she appeared to levitate rocks and throw them at Queen Beryl.
Her costume was a great deal fancier than a normal sailor fuku; everything was trimmed with lace, pearls, or both. Her chest ribbon was a light pink, with a large heart-shaped brooch at the center. Her skirt was still blue but had lacy white petticoats under it. The red gems on her odango were pearl-trimmed, and between them was a pearl crown similar to the one worn by Neo-Queen Serenity in the manga, in addition to the usual tiara. She wore white boots with red trim at the top in an upwards "v" shape and transparent heels. She also wore a ring on her left hand.
This costume was most likely designed by Naoko Takeuchi, as she released her own drawings of this character in official materials, such as program pamphlets for the musicals.