As the Internet gains users, the professionalism of your website become more important. Would you name your business "Susan's subdivision of the Bob Company"? No. So why be www.somebody-else.com/~your-company?
In addition, if you want to advertise your website in brochures, your letterhead, business cards, or other forms of advertising, your website address needs to be easy to remember. A certain percentage of your hit traffic will come through search engines and other links where the user doesn't have to type anything, but if you want people to be able to just sit down at their browser and go straight to your website, a domain name makes the job easier for them.
It's possible that the domain name you want has already been registered by someone else. Are you out of luck? No. In the near future, seven new domain endings will be added to the now-familiar .com, .net, and .org: '.store' for businesses offering goods, '.info' for information services, '.nom' for individuals who want personal sites, '.firm' for businesses or firms, '.web' for entities emphasizing the Web, '.arts' for cultural groups and 'rec' for recreational or entertainment activities.