Incubators provide space and other non-tangible resources. Oftentimes this means mentoring, services, and contacts.
Accelerators can provide what incubators provide, but most critically provide money for equity. The central reason why that happens is that accelerators believe that economic and equity alignment is necessary for all stakeholders to be incentivized.
Many people get these two definitions mixed up. But the one critical delineator is the conveyance of equity for money.
The meaning of these two terms, however, needs to progress. Incubators nor accelerators should not find it ok that they are labeled either one of these things for a duration of time. Incubators or accelerators are essentially startups to begin with, and startups must iterate over time or die.
Startups like Google, Apple, and Facebook are not what they were in the beginning. Google came from being a search engine company to now hardware. Apple was only a desktop manufacturing company but now mobile devices. Facebook was a copy of "Hot or Not" or "MySpace" but has now become a media company and even hardware.
The correct name that really should matter is to be called a startup. The moment an accelerator or incubator ceases to be like a startup and reinvent itself relentlessly, it becomes irrelevant. The moment it becomes a corporation, it becomes brittle and not innovative.