While casual travelers may (and do) charter private aircraft for transportation, the choice is more typically based on questions of time versus money. Chartering a private aircraft if by no means inexpensive, but while money may be in abundant supply, time is a finite resource. For example, if $10,000 spent on a charter flight results in closing a deal in which time is of the essence, and that deal is worth several million, than the $10,000 charter fee is money well spent.
This is perhaps the primary benefit of chartering a private aircraft. Whereas with commercial airlines you must show up two hours in advance of your departure time, stand in line for security checks then stand in line again for check-in, that is two hours that might have been better spent planning and preparing for that important meeting. In addition, commercial airlines are severely limited as to the airports to which they have access. In fact, altogether the commercial airlines of the world have access to no more than 500 airports across the globe. It is also highly improbable that any given airline with have direct, non-stop flights from any given airport to any specific destination. If you live in Los Angeles and need to fly to New York - or you're going from Seattle to Tokyo - then you won't have many difficulties, albeit on the airline's schedule - which may or may not be convenient for you. On the other hand, if you live in a smaller community such as Yakima Washington, Ogden Utah or even Rochester New York, you'll be forced to take connecting flights that can add time and inconvenience to your itinerary. (We won't even go into the issue of lost and misdirected luggage.)
A privately chartered aircraft can take you to any of over 13,500 airports and fields worldwide - directly from any location you choose. Since the companies that provide charter flight services are under private ownership, they are exempt from many burdensome regulations to which commercial carriers are subject. This is not to imply that safety is compromised; quite the contrary. This freedom from regulation allows charter companies to be far more discriminating in their hiring practices, being able to conduct the most rigorous background checks on all personnel. In addition, many private charter companies provide their own trained security and screeners. Since these people are dealing with far smaller numbers of people, they are able to be much more thorough in their jobs - meaning greater security and peace of mind for you as the traveler.
Private air travel is not for everyone. If you are among those for whom time is money however, it can very well be the most economical choice for all your travel requirements.