My guess is "no", at least as regards a classification. A check with GAP shows that 30,591 of the 34,297 groups of order $5^7$ satisfy this equality; that 7,882 of the 9,310 groups of order $3^7$ satisfy the equality; and that 53,499 of the 56,092 groups of order $2^8$ satisfy the equality.
Here's an easy bit of GAP code to check this:
glist:=AllSmallGroups(Size,5^7,G->DerivedLength(G) = 1 + LogInt(
NilpotencyClassOfGroup(G) , 2));;
Partial information may be attainable, such as constraining various aspects of their character theory, but off-hand I'm not sure. This paper of Glasby's seems to contain all the references needed for the original derivations of the inequality in question (he references ,, and  in his bibliography).