That's a Fred Brooks distinction, and not one that's (AFAIK) used outside of software consulting literature. In his essay, he basically uses "accidental complexity" to mean complexity from the past and "essential complexity" to mean "current complexity". He also says all software complexity is essential:

> The complexity of software is in essential property, not an accidental one. Hence descriptions of a software entity that abstract away its complexity often abstract away its essence.

This is pretty close to what I was saying.

There are, of course, more standard notions like Kolmogorov complexity. You could argue that the essential complexity of something is its Kolmogorov complexity. But I don't think anyone means that when they talk about essential complexity.