What's the difference between private practice and normal hospital?

The "hospital" doctors are employees of the hospital. A private practice doctor is employed by a private medical firm, but has privileges to treat her own patients in the hospital. So, if I'm in private practice and my regular patient has a heart attack and goes to the hospital, he can elect to have me treat him there instead of the regular hospital doctors.

One thing that was unrealistic about Elliott's private practice gig in Scrubs was that she was at the hospital about 100% of the time and still answered to Cox and Kelso like they were here bosses. In real life, she would have spent about 50% of her time in her firm's private medical office, and spent the remaining 50% split among several hospitals wherever her firm's patients might be. And Cox and Kelso wouldn't have been bosses to her anymore.

You should watch the entire Grey's Anatomy series to date and then watch the entire series of Private Practice on Netflix  while keeping in mind that they are both fictional but that should clear it up for you

In actual fact, most physicians in the US are not "employees" of hospitals (unless they hold administrative positions) - they are all in private practice and have admitting privileges.

In the last couple of decades a new specialty has emerged - hospitalists - and in some states they may actually be employed by hospitals - in my state (Texas) it is avtially illegal for anyone other then another physician to *employ* a physician so even the hospitalists are self-employed or are employed by a physician group.

http://www.diabeticlifestyle.com/eating-well/diabetic-food-exchanges-made-easy

But, this is fiction so they're there to tell a story, not describe the US hospital system.