Church Latin and Classical Latin were quite different phonologically. Classical Latin was syllable-timed and had vowel length and predictable stress; Church (i.e, Medieval) Latin was stress-timed and had lost vowel length and stress was (therefore) unpredictable. My guess is that if a fluent speaker of one and a fluent speaker of the other had to discuss something they both knew about, they would each think the other's accent barbarous, but they would eventually be able to understand one another.
The issue is native speakers; there were many fluent speakers of one dialect or another of Medieval Latin over its 1500 or so years, but no native speakers born in a speech community. For the first 15 centuries of the Christian Era, anyone in Europe who read or wrote anything -- with very few exceptions -- read and wrote in Latin, whatever language they spoke natively. Thus, all fluent Latin speakers were literate, having learned the language in order to be able to read and write. This constituted a very small (but very important) proportion of the population. However, this is the case for Classical Latin as well.
By the time of J. Caesar, the language spoken in the streets of Rome was not the same as Classical Latin, since it was Vulgar Latin -- a quite different language, and busy turning into all the Romance languages (Modern Italian is basically the Tuscan dialect of Vulgar Latin with two millennia of age on it). The first thing that happened was a lot of sound changes, leading to a total restructuring of the grammar -- no cases and only two genders, for instance. So the fluent speakers of Classical Latin from early Imperial times were not native speakers of it, either. Diglossia was a prominent feature of European language communities, and still is in many places.
So, since the language was identical when written for 1500 years and could be spoken by many at any time (it was necessary for any serious work), though the pronunciations and word meanings changed, as necessary, when the cultures did. I'd say they were the same language, as much the same as Mexican Spanish and Peninsular Spanish, say.