To answer your last question directly: No, private companies do not have to be concerned with SOX.
Other differences may include whether your financial statement is audited or just reviewed by an accountant; the relationship the BOD (Board of Directors) has to the company; as well as the internal procedures the company establishes.
I am currently a private held company whose parent corporation is in the process of going public. The number of changes that we have had to go through in the past two years is absolutely incredible. I went from being reviewed by one auditor at the end of every calendar year, to being audited every quarter! Thankfully, I don't have to worry about SOX as of yet (my parent corporation is in Taiwan), but I'm not that far away from being compliant.