I heard today (from a friend, please enlighten me if this is not the case) that Church Latin has/had distinct differences in pronunciation and/or grammar than the Latin spoken by the Roman populace. Would the Romans have understood Church Latin?
You mean, say, Romans under the reign of Augustus? Yes, absolutely.
There are some phonetic changes - mainly, hard 'c' (k) softens to 'ch'. There are also some grammatical changes, but nothing that would remain incomprehensible.
They would have significantly more difficulty understanding the Latin commonly spoken from ca. 550 to ca. 800, which was proto-Romance.
There are no good English books on the development of later Latin, but if you can read French: