Am I morally obliged to report what I judge as self-plagiarism? I tried to search in the previous Q/As, but I do not think my case falls in any of the previous entries.

While looking for a certain topic in my field, I found two papers. 
The first paper was authored by authors A, B, C, and D and published in some conference proceedings. The second paper was published in a journal in a year after by authors A, E, D, and C. That is, the second paper lost one author and gained a new author.

The two papers report the same research hypotheses, the same data collected, the same data analysis, the same results. That is, the "numbers" in the papers are the same (even the table contents).

The texts differ in a subtle way. The first paper is entitled "Studying topic T in the domain D". The second paper is entitled "Using Measurement Instrument M to measure T in the domain D". The second paper somehow avoids copy/pasting of the paragraphs from the first paper-they rewrote the sentences. It is a little bit longer than the first one. The authors actually spent some more words talking about measurement instrument M, which is taken from another discipline and it takes 3 minutes of Google Scholar searching for learning more about it. There is even a Wikipedia page about the instrument.

I think that this is a case of self-plagiarism. The second paper does not add anything useful to the first one. Additionally, the second paper does not cite the first one. This ringed a bell.

Please note that in my field, conferences are archived and are often considered more important than journals. Regardless of which venues are more important, both papers are formally considered publications.

Now, am I morally obliged to contact the editor of the journal where the second paper was published? This is not precisely a double publication, nor are the authors damaging me nor anybody else. Still, I feel they did something wrong. Should I instead live and let die?
This is an interesting case that you cite. I think it falls somewhere between 'standard behavior' and 'clear misconduct' (though I am sure others will disagree). Some thoughts and facts:


In Computer Science, even though both journals and conf proceedings count as papers, it is common accepted practice to submit extended versions of proceedings papers to journals. Typically, journals expect between 30% and 50% new content for such papers.
Realistically, many journals (especially ones that are not exactly top quality) are not super-strict with what they consider "additional content". Describing only their methodology in more detail, without providing new insights or results, is certainly very little new contribution, but I am not sure if that already qualifies as misconduct.
That one of the original authors was missing in the journal version is curious, but there may be reasons for that. Presumably, something a little shady has happened along the lines, but it is impossible to tell from the outside. One realistic possibility is that first author A has submitted the conf paper while at institution (1), with help from B and C. D was the supervisor and contributed little to nothing at all, but was added due to lab policy. Now, before submitting the journal version, A moved to institution (2), and has replaced lab head D with lab head E.
What strikes me as very, very odd is that the journal paper does not make clear that it is an extension of an earlier conference paper by citing it (and saying so explicitly in the introduction). This is the one thing that I certainly agree has a very bad smell to it. If I receive such a "hidden" journal extension for review, it is an auto-reject for me. 



  Now, am I morally obliged to contact the editor of the journal where the second paper was published? This is not precisely a double publication, nor are the authors damaging me nor anybody else. Still, I feel they did something wrong. Should I instead live and let die?


In summary, I would say the entire case certainly does not look good. That being said, I do not think that the case is blatant enough that you need to feel required to report to the journal editor. To be honest, even if you did, I think not much would happen. Likely, it is not a paper that the journal feels overly proud of, but I do not think that the case is bad enough to warrant formally retracting the paper.