It seems to me (although I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of non-commutative algebraic geometry refers mainly to non-commutative spaces represented by abelian categories, while non-commutative geometry derived from Kontsevich, Orlov, Efimov, Toen, etc., studies more general spaces represented by improved triangulated categories (DG categories, $ A _ {infty} $-categories, stable $ infty $-categories etc.)

However, it seems that this is not the case: Rosenberg introduced his notion of a spectrum (left) not only for abelian categories, but also for triangulated categories (this corresponding derived geometry at the level of generality) and to correct the exact categories with weakness equivalences (which, apparently, is a generalization of Barwick's notion of an exact $ infty $-category where "accuracy" is unilateral (see here and here).

Apparently, the key difference is the notion of a spectrum of an underlying noncommutative space: somehow, derived noncommutative geometrists do without it. It seems that the presence of spectrum makes the idea that the category is the "category of quasi-sheaves in a non-commutative space" more rigorous in a category, since a non-commutative space is represented by a categorical analog of a ringed space locally, that is, a stack of certain categories.

I would like to know more about the differences and similarities between these two approaches. In particular

What does the spectrum do in Rosenberg's noncommutative geometry and how does the geometry derived from Kontsevich manage without it? Would it be possible to introduce the notion of spectrum into derived noncommutative geometry?