There's really no difference between them, as they are currently being used:
Meanwhile, almost all questions tagged network-flow are actually about finding a maximum flow.
While in principle it would be possible to create a meaningful distinction between them, (a) it'd be a very fine line, (b) no such distinction currently exists, (c) people aren't actually using the tag that way, (d) we don't have any tag wikis or anything to guide posters to use the tags that way and most posters probably don't read tag wikis anyway, so posters will continue to use tags in a way that does not respect those fine distinctions, (e) the two are so close that I don't see much value in drawing that particular distinction anyway.
I would vote to keep max-flow. Max-flow should apply to a strict subset of network-flow questions. We also have shortest-path, which is another class of network-flow algorithms. It has 151 uses, which is more than double what we have on network-flow.
Shortest path problems can be viewed as specializations of the minimum cost flow problem with costs but no capacity constraints on the links. Similarly Max-flow problems can be viewed as specializations of the minimum cost flow problem with capacity constraints but zero costs on all the links.
Similarly the min-cost flow problem is itself a specialization of the linear programming problem, and multi-commodity flow problems can be formulated as integer programming problems, but I don't think we want to get rid of the network-flow tag just because we already have an integer-programming tag.