Nothing that will affect most users. The main change is that (1) the MathOptInterface support is now a package extension, so that interface (used e.g. for JuMP) only gets loaded if MathOptInterface is loaded (either directly or by another package). Also (2) the obsolete MathProgBase interface was dropped.
(Realize that even going from 0.6.5 to 0.7.0 would be considered breaking in semver. See also the discussion in Encouraging 1.x versions for registered packages? - #56 by Eben60 — going to 1.x.x makes sense because the main NLopt interface has been stable for a while, and major versions ≥ 1 give strictly more information in semver.)