Status Symbols
In a universe where a majority of the characters wear uniforms, it becomes necessary to include identifying features or accessories to liven things up. Byakuya dresses up his hair, neck, and hands to complement his black shihakushou and white captain's haori, mostly as a result of his noble rank.
Hairpieces
The most prominent of the aristocrats in Soul Society wear ivory-colored kenseikan as a distinguishing symbol. They have an elongated, triangular shape, though it seems variations in how and where they are worn do exist. Byakuya's grandfather Ginrei actually wore no visible ones, while his father's were vertically clipped to the side. Byakuya himself had several pieces spread over two different areas of his hair, adding structure to his hairstyle and enhancing the sharpness of his silhouette.
Following the time jump, the iconic kenseikan seem to have been traded for a more subtle set of hairpieces that sit toward the back of Byakuya's head, allowing his hair a bit more freedom. There has been no indication of a symbolic meaning behind this change, but I do feel that it softens his look and matures him a little.
Neckwear
The neck area is a prominent part of the body considering its proximity to the face, so displaying decorations there would make them hard to miss. The heads of the Kuchiki family have passed down an elegant, inexplicably expensive scarf to their successors as a symbol of their position. It proves to be an integral part of Byakuya's image, so that even when he haori is absent, he is still identifiably noble.
In a flashback sequence, Byakuya wears a distinct, white haori (though he was not yet a captain), the collar of which was held together by a fringe-like necklace. Its significance is never explained, but there is no doubt it is yet another indication of his rank and gives him an overbearing aura.
Interestingly enough, the updated character designs has Byakuya in a new haori and scarf-less. An edited high collar features a gold tassel on either side, as well as gold lining on the seams. As with the change in headdress, the new neckline tightens up his silhouette without sacrificing his stateliness, but any symbolic meaning behind it has yet to be revealed.
Hand-coverings
More subtle parts to the 6th Division Captain's style are the white hand-coverings (tekkou) that protect the back of his hands and around his wrists. Very few characters were designed with anything around or on their hands, though it might not be coincidence that Ginrei was one of them.
In general, Byakuya is always well covered, to the point where he'll even sport a light hooded sweatshirt while at the beach. It'd be unbecoming for a noble to be so revealing with his body, so at least that's something he's modest about. Therefore, the hand coverings limit his exposure when in uniform, serving both aesthetic and defensive purposes.