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The articles on Wikipedia: Gibbs measure and Boltzmann distribution make it seem that there is a difference, in particular in the following sentence in the first link referring to Gibbs measure:
It is a generalization of the canonical ensemble to infinite systems
I think that the wikipedia article is going too far: the terminology "Gibbs measure" applies both to finite and infinite systems.
For finite systems, "Gibbs measure", "Gibbs distribution" or "Boltzmann distribution" are used interchangeably. For infinite systems, however, one indeed only uses the terminology "Gibbs measure". (In mathematical physics and probability theory, it seems to have become more common to use "Gibbs measure" in all contexts, reserving "Boltzmann weight" for the expression $e^{-\beta H}$.)
In addition, it is somewhat customary to say "finite-volume Gibbs measure" and "infinite-volume Gibbs measure" when one needs to stress the context.
Finally, let me also mention that the property that characterizes Gibbs measures (the fact that the DLR equations hold) applies both in the finite and infinite settings. It is however easier, in the finite setting, to define the Gibbs measure directly as being proportional to $e^{-\beta H}$.
