One of the joys of being wealthy is being able to buy your way out of the inconveniences that the rest of us face. Flying private means you don't have crowded airports, long waits, the plane leaves on your schedule, no baggage claim carousel, no crying babies on the flight, lots of leg space and comfortable chairs, good food/beverages, total privacy and tons of time saved.
You call the company and tell them your address and destination address and just wait for the limo to arrive. It'll take you to the airport where the plane will take you to your destination and a limo at the far end takes you to the building you want.
No security, no lines - car to plane in five minutes. It's a custom-made nonstop flight on the right plane for the job. It also gives you a lot more options for airports as the airlines only serve a fraction of the available airports. If you fly commercial, getting to places like Vail, Jackson Hole, or Majorca can take two or three flights. With the right aircraft, you may be able to fly non-stop from Sardy Field to London City.
Most people who travel privately do it mostly for domestic travel. Many people will fly privately from their home city into a bigger airport and then catch their international connection on a commercial flight. You can instruct for inflight catering and it can be anything you can imagine. If it's international, you have to wait for a customs agent to board the plane, check and stamp your passport - you're now cleared through customs, right on the aircraft.
If you are a wealthy or famous person, there may be people who dislike you or who have mental or emotional issues that prevent them from controlling their behavior while on an airplane. With private jets, you do not have to worry about that. You can ensure that everyone on the aircraft is an employee of the company under some NDA.