In the context of this article, I think you can safely exchange those two words. I'd even say you can exchange company by side-project and the advice is still solid.

It's probably because running a company usually means you have more resources and money available to use than just doing a project. I guess that it comes with a lot more constraints than a project though in some areas - you might have really good ideas that you want to try but if they're too big and too much of a gamble, you can't really just do it and risk your investor's money...

Equipment/software/subscription purchases are tax write-offs.

A company presumably has the relevant filings and registrations done, and tax and accounting processes set up, so if a project starts generating revenue or is ready to do so, that's all taken care of.

ie, you're not sitting there with an app ready to go in the app store, trying to decide which business structure to use, how to get it established, filing to get a tax ID #, etc.

Plus, write-offs.

A company has customers. Part of the value was that he could try out new products/features on an existing customer base.

That part resonated with me and after parting with my company was something I definitely missed.