What is the conceptual difference between Gibbs and Boltzmann entropies?

The Gibbs entropy is the generalization of the Boltzmann entropy holding for all systems, while the Boltzmann entropy is only the entropy if the system is in global thermodynamical equilibrium. Both are a measure for the microstates available to a system, but the Gibbs entropy does not require the system to be in a single, well-defined macrostate.

This is not hard to see: For a system that is with probability $p_i$ in a microstate, the Gibbs entropy is

$$ S_G = -k_B \sum_i p_i \ln(p_i)$$

and, in equilibrium, all microstates belonging to the equilibrium macrostate are equally likely, so, for $N$ states, we obtain with $p_i = \frac{1}{N}$

\begin{align} S_G &= -k_B \sum_i \frac{1}{N} \ln\left(\frac{1}{N}\right) \\&= -k_B N \frac{1}{N} \ln\left(\frac{1}{N}\right) \\ &= k_B \ln(N)\end{align}

by the properties of the logarithm, where the latter term is the Boltzmann entropy for a system with $N$ microstates.