First of all, I don’t think there is a clear categorical line to draw here. Complexity and complex networks are nothing clearly defined (see this answer of mine). In particular social networks are now investigated by “complex-network people”, so they are complex networks as well.
I think the best way to understand this is historically:
Around the year 2000, physicists (and others) began applying network methodology from sociology (and other fields with straightforward networks) to other networks, such as neural networks, gene-regulation networks, and dynamical networks, i.e., theoretical systems of coupled oscillators. As a result, a considerable amount of scientists from all fields suddenly became interested in networks and brought new perspectives into the field, which resulted in new methods and insights (and, to be frank, a lot of vanity research).
One crucial example of this is the introduction of new network models, such as Watts’ and Strogatz’s small world and Albert’s and Barabási’s scale-free networks. From the physics perspective, the new thing was that people investigated the dynamics happening on these networks and it turned out that they were different from the more simple networks physicists primarily had considered so far, namely lattices, complete networks (mean field) and maybe random networks. The prior focus on simple networks can be attributed to them being more relevant in orthodox physics (e.g., crystals), allowing for theoretical approaches, and being easier to simulate (which was much more difficult in those times).
So, in brief, what happened is that physicists took existing concepts from sociology and generalised them. This generalisation brought new and different people to the topic and thus resulted in new applications and methodology. Had the general approach come first (say, in 1900), you probably wouldn’t consider a divide like this.
Sidenote: Of the two points you listed, the use of statistical physics is not surprising in this light anymore, while the amount of data is simply due to the fact that some (but not all) of the new fields of application had bigger datasets available.