The terms are used interchangeably, but the two are not the same. In fact, they’re very different.
Both offer office space; incubators allow you to stay longer.
The keyword for accelerator is ‘action.’ For incubators: ‘function.’
Accelerators have a stake in your success, and will do everything they can to see to it that you are successful, even after you’ve been through the initial program (many entrepreneurs are not aware of that): an investor is an investor, and a good accelerator is invested in you from start to eventual exit.
New York has a plethora of coworking spaces (which can also offer help, depending on who else is in the space, and some investors hold office hours there), several accelerator programs and a few incubators, NYU Poly, inSite and WeWork Labs being among the better known.
Companies that come out of incubators have been known to get funded: a good company is a good company, no matter where their offices happen to be.
Reddit, dropbox, sendgrid, taskrabbit and airbnb all came out of accelerators, and this that’s a very abbreviated list.
Are these programs critical to your success? Not necessarily, but they can and do help you to turn what your concept into a viable business, complete with value proposition and a path to revenue, clearly stated. Accelerator programs are an excellent option for early stage entrepreneurs to consider because they offer a vast array of benefits that will literally propel your startup forward. They offer classes to help you get your business investor-ready, and industry/subject matter experts as mentors are available to help when you get stuck.
Investors also pay attention to companies that go through these programs: they’ve been pre-vetted. Then, come Demo Day, the investors come to you.
Starting any entrepreneurial endeavor is like jumping out of an airplane before you know that the parachute fully works. Something can always go wrong. With accelerators, good to know there’s an instructor there, and you know he/she’s got your back.