Not alot really imo, it's mostly personal preference.
With the VMJ you get a maple, black blocked bound neck, although the blocks are painted on and arn't real, and the neck has a rather lovely satin finish to it. I've got one, it's alright, I want new pups for it as they're not what I'm looking for soundwise. And the body is all maple.
The Classic Vibe gives you a synthetic bone nut (same one as the MIA Fenders I believe), a High Mass Bridge (much better than the stock bent tin that's on the VMJ, again the same as the MIA Fenders) and you get vintage style frets (slightly thinner fretwire methinks) and a basswood body coupled to a gloss finished neck sporting a rosewood fretboard. The basswood will be lighter and easier to wear for long periods, my MIJ Aerodyne is basswood, nice and light.
They're both the top of the Squiers so QC shouldn't be a problem.
Personally I'd go with the Classic Vibe if I had to replace mine, partly as it seems to come with newer and better designed hardware, but also because the price difference and I'd quite fancy a white jazz with a tort plate. Don't get me wrong, I do like my VMJ, it's nice to play and definitely worth the £230ish it cost me 2 summers ago but I think I'd rather have the better nut.