In the first Hellraiser movie, the opening scene shows Frank Cotton solving the puzzle box that calls forth the Cenobites and immediately being torn apart by the hooks. Later in that same movie, Kirsty Cotton opens the box and, instead of meeting the same fate, inadvertently calls the Cenobites and exchanges several lines of dialogue. In the book, The Hellbound Heart, solving the box just summons the Cenobites, but they give Frank a choice as to whether or not to go with them. However, when Kirsty solves the box, they tell her that she MUST go with them. I haven't read the later books yet, so is it ever explained why the Cenobites take their time to claim some victims, but instantly kill others, or why one had a choice as to whether to go with them, while another didn't (Until she bargained for her life)?
You are asking about two different works, a movie and a book.
With that being said, there is no explicit explanation in either canon. My best guess would be something about desire. In both works Frank knows the box supposedly leads to something and he wants whatever it is.
Kirsty had no clue what the box actually was. She stole it cause it looked important.
As to choice, the offering of a choice or not is in itself potentially part of the torture. Frank is the type who would ultimately regret the choice he made while not offering Kirsty a choice would be more torturous to her.