When NOT to use Publishing & Communication Sites?
If you have a scenario where few people (generally one or two) are creating content for a larger audience (generally whole organization) then use publishing sites or more recommended communication sites.
Team Sites. Do not use Publishing Sites or Communication Sites for such scenarios. Its an over kill. Also between small groups the communication is generally two way.
Publishing Sites vs Communication Sites:
Now that we have learned the major differences between the both site types, here is what we should do in order to choose between them.
We should start designing publishing solutions with Communication sites in mind. It gives many features OOB which have been requested by customer again and again, specially the responsive UI. It is quicker to create content, faster to publish and cleaner to manage Communication sites without any developer's involvement. At this point it does not have features like cross-site publishing, Approval workflow, multi language etc but many(not all) of these features are in pipeline as told in community forum :o).
Note: Publishing Sites are not deprecated at this point.
On the other hand Publishing Site have more functionality at this point and gives you more control over branding as you can introduce your own HTML/CSS and can write custom MasterPages. But this great functionality comes at the cost of custom code and developer's involvement which can result in higher cost for publishing.