I came across the question "Attitudes towards self-plagiarism" on this site, which asks whether it is OK to reuse text from your own papers without quotation formatting. It seems the answer is no. I didn't know this and I have in fact done this in the past, before I even heard the term "self-plagiarism."
Here is the precise situation: I wrote and published paper A. Then I kept working in the same area and I extended the results, so I decided to write paper B two years later. When writing paper B, I started by taking paper A and editing it. Consequently, much of the text in paper B is directly copied from paper A. In particular, many paragraphs in the introduction, related work, and background sections are either identical or lightly edited. There is no indication which sentences are copied and which are not.
Of course, paper B clearly explains the difference between the results of paper B and the results of paper A. In particular, paper B solves an open problem posed in paper A. As far as content goes, there is no problem, in the sense that, if I had rewritten the copied parts, there would be no problem.
When I did this, I didn't know it was considered bad practise. I figured that, since the motivation, related work, and background for both papers is essentially the same, there was no point in writing paper B from scratch, given that paper A had already covered these things.
What should I do about this (if anything)? I don't think I did anything malicious, but it seems this is not acceptable.
In paper B, if you cited paper A, the paper from which the major content of paper B is from, then your act would not completely be pictured as self-plagiarism. This is because plagiarism is an act of inclusion of matter without attributing the original source.
There is also another case of duplicate publication, which also doesn't apply to your condition as you stated that you've improved the method and differentiated the results.
Final verdict: Provide the methods and results are clearly differentiated, there is no need to worry. You haven't done anything seriously unethical to be concerned too much about. If this still bothers you, then you may suggest for a revision of your paper to the publisher of paper B to make necessary changes in the related work and background section.