I think the answers above pretty much explain the basics. From my personal experience, here are the two instances where I decided to join a private club:
1.) A few years after graduating from college, I was feeling like I needed a new challenge to keep me motivated in tennis. There happened to be one private club in town that had clay courts, and I had only played on clay a couple times. They were running a special where there was no initiation fee, so I joined primarily to have access to the clay and learn to play on that surface. I think I was there for about 18 months. I went from getting my butt handed to me on the clay to finally getting wins over the two best players in the club on those courts. I never learned to slide properly, but got used to the bad bounces and extra shots you have to hit on clay. I also played on the club's USTA league team, used the workout room, and took an occassional fitness class. I ended up leaving because I got burned out on tennis again and my job changed.
2.) I joined my current private club because the public facility that I was playing out of changed the monthly rate of their family memberships to $90. That may seem like a steal, however, getting court time was nearly impossible because the memberships were oversold and they squeezed too many lesson groups (primarily for juniors) into the prime court times. The front desk staff was also indifferent or just rude/idiotic at times, and the best players only liked to practice in their own little group. The public facility also only had tennis and racquetball courts, so there was no pool or workout facilities. The private club I joined has two pools, a jacuzzi, sauna, steam room, two workout gyms, basketball and racquetball courts, golf, and of course tennis. The family membership is $150 per month, and they had a special where the initiation was only $50. I'm spending $60 more a month, but my whole family is enjoying all the features, so it's a much better deal.
Like all gym-type memberships, if you are actually going to use the features of the facility 4-7 times per week, then it's well worth it to join. However, if you are only doing sporadic activity, you'd be better off doing "pay per play" or free in public facilities.