Supernaturals and ghosts are both used interchangeably and share many traits with each other. In other words, ghosts can also be referred to be a type of supernatural. Both need rumors in order to exist on the Near Shore, and can either originate from the Far Shore themselves or created out of someone's ideals in special cases.
Unlike ghosts, they can show themselves at will to normal humans. Despite being supernaturals themselves, Tsuchigomori can disguise himself as a teacher by hiding his spider-like appendages and long ears, in addition to Yako being seen by Misaki in her human-like and fox appearances (including his students catching glimpses of her and assuming the courtyard must be haunted). They are able to be touched and can interact with ghosts and humans. They are presumably able to age unlike ghosts, as signified by Tsuchigomori's older appearance and Yako's growth from being a child to her current adult like look.
Many supernaturals have certain abilities special to them. However, they can also be modified by rumors before being reverted back to their original form. Depending on how popular a rumor is or how much power they are given (eg. supernaturals like the Mokke comically aiming to gain a School Wonder seat), their form will change no matter how dangerous or powerful they will become.