Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about $\mathcal{D}^{b}(\mathrm{Rep}\ kQ)$, i.e. derived categories for representations of quivers, and am going on later to talk about $\mathcal{D}^{b}(\mathrm{Coh}\ X)$, i.e. coherent sheaves.

Question: I want to motivate studying $\mathcal{D}^{b}(\mathrm{Coh}\ X)$ by saying, analogously to the representation-theoretic setting, "derived equivalence is good because it's not too strong but it still detects key properties such as..." - and here, I need some help. What properties of a variety are detected by the derived category of coherent sheaves?

I'm aware of the Bondal-Orlov reconstruction theorem, which is obviously as good as one could hope for but doesn't always hold. But one expects some invariants to be picked up by the derived category, especially cohomological ones, but are there others too? Of course I'm deliberately being vague about what I mean by "variety" here, as I expect the answer to vary depending on what one asks for.

Disclaimers: IANAAG (as they say), as you'll have guessed from the question. I'm aware of this question on MSE and this one on MO and a few other similar ones but I feel that none tackle my particular query - if I missed one that does, mea culpa. I also know of some survey-type articles on the topic by various authors, where one might have expected to find this question addressed, but I haven't seen it. I'd be very open to a good reference in lieu of an answer. I also don't see a community wiki button but surely this should be, if the powers that be could make it so, please.