Generally, at some point an Abbey must have housed a community of monks headed by an abbot, while a Church serves solely as a place of worship for a community of all sorts of people.

I get the impression that England, France, and Germany, having been comparatively less urbanized in late antiquity, there are more rural religious communities (ergo, abbeys) than in a more urbanized region, like Italy. It's also worth pointing out that there are several notable cases of abbeys which had at one point been located in suburban or rural locations getting engulfed either by the growth of their own community or the growth nearby community, and consequentially the role of the abbey in question might change; the most notable case probably being Westminster Abbey in London.

I wrote about abbeys in the context of Italy here, although I think you might be more interested in the English context, which I'm not exactly an expert on!