Which is better really depends on the situation. Also, sometimes it is not a choice that is up to you, but the work type and relationship dictates whether an employee or independent contractor is the correct legal relationship.
Here is a good article on the subject
Eric is correct in pointing out the legal differences which may enter into your decision.
I was once an independent contractor which was my preferred way of operating. From my perspective, the advantages were that I could set my own rates and dealt with my clients directly without a middleman. I did things the way I wanted them to be and was not forced into procedures dictated by a club. There were other trainers around me, some employees who also worked shifts at the club, other were independent trainers. I paid the club for the use of the equipment and had an understanding on how to train members vs non-members.
Of course, this mode meant that there was no homogeneity in training procedures within the club but since we were all identifiable as independent trainers, that did not seem to matter to anybody. Since I paid a fee per person trained, it was on the honor system, and I for my part was scrupulous of paying the club its dues. I have no idea how the other trainers handled this. I was also responsible for my own insurance (no big deal, I needed that anyway) and had to find my own clients because new members to the club were usually ‘assigned’ to a trainer on staff if they inquired into personal training.
I felt that independent contractors were an asset. There were often there and always willing to answer member questions which meant that they were assisting without being paid as staff. The variety of trainers with different backgrounds made the question of what to pay experienced trainer vs novices irrelevant because we were setting our own rates.
So much for my opinion ….