If I recall correctly, the classical reached it's zenith (with Cicero) around the first century prior to the collapse of the Roman empire and it wasn't until about the 4th century with the creation of the Vulgate bible that ecclesiastical Latin took over as the primary spoken form. Other than small portions of "new" vocabulary in the living Latin of the ecclesiastical branch, the primary difference between them now is in pronunciation. In most pedagogical settings, it's the classical Latin which is taught as it's commonly considered the most refined (since the Florentine Renaissance anyway) and typically it's writers from that period which are focused on most rather than church fathers of the later years like Augustine, Tertullius, Ambrose, Jerome, or Ignatius. Naturally, this is a highly simplified viewpoint and there are philologists who would subdivide the language in much finer gradations (Golden age: Republican, Augustan; Silver Age; etc.)
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