What's the difference between these two words, abbreviation and abbreviature?
"Abbreviature" appears to have two purposes: an outdated synonym of "abbreviation" and an entirely additional meaning:
An abridgment; a compendium or abstract.
"This is an excellent abbreviature of the whole duty of a Christian. - Jeremy Taylor"
So the current difference is that "abbreviature" applies to large bodies of works or an entire subject; "abbreviation" is more applicable to words and phrases.
Abbreviation is a common English word which normally means a shortened form of a written word or phrase.
Abbreviature is a far rarer word (somewhat technical and/or archaic, in my opinion) that normally means a condensed version of something significantly longer than just a word or phrase. An abstract, summary, or digest of an instruction manual, for example.