I suspect that these phrases are usually used in different contexts. To me, X is not consistent with Y usually means that either X is false or Y is false; but X is not in keeping with Y means that X does not accord with the principle, mores etc of Y.
[verb phrase] to not seem right or suitable when seen together with something else that is very different in style or character
Serious poetry published in newspapers always seems slightly out of keeping.
be out of keeping/not be in keeping with
Those new windows are not really in keeping with the style of the house.
The cheerful cover of the diary was out of keeping with the thoughts I was recording inside.
Longman Dictionary of Contemporary English
Also I think that in the first example from my dictionary not consistent with wouldn't work (but please correct me someone if I'm wrong).