Is it the size of the vision? (ie, a boutique cafe will only ever be small, while a start-up might be valued at $15B) Or is it the difference between leveraging effort rather than leveraging scalable code?
To be a startup, a company has to be a product business, not a service business. By which I mean not that it has to make something physical, but that it has to have one thing it sells to many people, rather than doing custom work for individual clients. Custom work doesn't scale. To be a startup you need to be the band that sells a million copies of a song, not the band that makes money by playing at individual weddings and bar mitzvahs. -- http://www.paulgraham.com/startupfunding.html
Likewise, for startups, the distinction isn't always that cut and dry. If you're an Oracle or a Salesforce.com (both of which were once startups) you probably had to do customization for every one of your first n clients in order to make the sale.
I think there are obviously businesses that let you scale easily, and ones that don't, and the quote above is a good starting point, but it's not a 100% clear line. And often what looks like a scalable business turns out not to be so, and vice-versa.
Edit: Reading the responses to this, people clearly got the wrong idea. A liquidity event != early acquisition. But every successful startup nonetheless hits liquidity: acquisition, IPO, or substantial dividend payment to its shareholders. And that's a distinction between a successful startup and a successful small business, which are not intended to achieve liquidity.
A "liquidity event" (god I hate that term) might be possible but it's not the driving force. If someone offered me a huge sum of money, I might be tempted, but I'd just take that money and work on the next thing I considered needs to exist!
I don't think people obsessed with the goal of money beyond anything else create startups, although they might find great riches as a result of their efforts. If money was the only thing in my life I'd probably have tried to found an investment fund or a real estate company or something.
Sorry to talk about myself here but in such things one can only really refer to one's own experience.