What is a start-up? How is it different from any other new business?
The best way to think about it is that when a business is formed, it's new, and funding can take shape in almost any way. A start-up is a sub-type of new business, and when it happens, the type of investment is fairly narrowly defined, and the expectations for success and failure are much more dramatic.
There's a great post over at I, Cringely about, oh, Innovation (yes, with a capital 'I'), I guess. Within it is a very clear description of the difference between a standard-issue new business, and a startup.
Knowing which one you are in, or about to get into, and how to describe it to different audiences may well be the fulcrum on which your future success rests.