I believe there are two distinctions to be made: the difference between a photo-story and a photo-essay, and the difference between and insider and outsider view.
Both a photo-essay and a photo-story are what David Hurn (quoted in course notes p52) describes as ‘a group of images in which each picture is supporting and strengthening all the others’. The difference seems to be that a story has a distinct narrative flow but an essay is more free-form, but I suspect that the best projects have elements of both. The Dad Project is primarily a story (charting the progress of Dad’s illness, decline and death) but with non-linear elements. Country Doctor is primarily an essay but has four mini-stories embedded in it.
Abigail Solomon-Godeau (1994) discusses the difference between documentary shot by an outsider (more objective but less ‘involved’) and an insider (probably more detailed knowledge, but a more subjective approach). Campbell was pretty much the ultimate ‘insider’ to the extent that she was, herself, part of the story she is telling. Smith was mostly an outsider; his essay appears more objective – although he clearly has an admiration for his subject.
The other difference between the two series is their historical context. Country Doctor appeared in the late 1940s, shortly after the Second World War and with the Depression well within living memory. Many of his readers will have lived through hardship and approached it with the kind of stoicism and sense of duty projected by Dr Ceriani. The Dad Project is very much a product of the early 21st century, when it is OK to be open about emotions in public.
Campbell, B. (n.d.). The Dad Project. [online] Brionycampbell.com. Available at: http://www.brionycampbell.com/projects/the-dad-project/ [Accessed 10 Apr. 2017].
Cosgrove, B. (2012). W. Eugene Smith’s Landmark Portrait: ‘Country Doctor’. [online] Time.com. Available at: http://time.com/3456085/w-eugene-smiths-landmark-photo-essay-country-doctor/ [Accessed 3 Apr. 2017].
Solomon-Godeau, A. (1994). Inside Out. [online] Available at: http://www.photopedagogy.com/uploads/5/0/0/9/50097419/week_5_abigail_solomon-godeau_inside_out.pdf [Accessed 4 Mar. 2017].