Right now, we don't really describe what a social network is at all and online community just says "An online community is a web site providing information, music, and/or news, and may act as an unofficial homepage for an entity." which is rather vague and covers all sorts of things which aren't communities.
At some point I was working on some descriptions for online community (https://musicbrainz.org/relationship/35b3a50f-bf0e-4309-a3b4-58eeed8cee6a ) and social network (https://musicbrainz.org/relationship/99429741-f3f6-484b-84f8-23af51991770 ) which I was pretty happy with, but unfortunately I still can't find where on earth I saved it, so I'll just have to start again...
My proposal would be to describe them along the lines of the following:
An online community is a group where any member of the group can comment and contribute content. The artist may or may not be a member of the group. Examples include forums, LiveJournal communities and groups on Facebook, Last.fm and Flickr.
A social network page is an artist's own profile page on a social network (https://en.wikipedia.org/wiki/Social_networking_service) which only they can post content to. Other people can create their own profiles and interact with the artist, e.g. by adding them as a friend or by commenting on the things that they post. Examples include Facebook user pages, Last.fm users and accounts on Twitter, Instagram and Flickr.
The main problem I see here is that we currently use "social network" for Last.fm /music/ pages (also label, venue and event), but it doesn't fit under either of these descriptions. I think the best thing there would be to move Last.fm URLs (other than the user and group ones which are covered by the descriptions above) to be under "other databases", either using "other databases" directly or adding a new relationship type like we did for Songkick (note: if we do this, remember that quite a few releases have download for free relationships for Last.fm).
The other thing I would change is that, right now, for artists, online community acts as the parent for a bunch of relationship types, including social network. For other entities, those relationships are grouped under "online data". I think we should make artists consistent with the other entities and have both online community and social network under "online data".