Dieselpunk is basically considered a variant of steampunk rather than something fundamentally separate. Cyberpunk was a term bandied about in the early 1980's which was then applied to the works of a whole group of authors, predominantly William Gibson and Bruce Sterling, as a literary movement of its time with the works having similar thematic and stylistic material -- young people, alienation through technology, rebellion, noir suspense, etc. Such works had been done before and were also part of the New Wave literary movement in SF, but the term referred to a specific group of authors deliberately exploring the same territory in the same stylistic and thematic approach -- punk (as in the music/lifestyle/art) with technology and noir. The term was later more broadly applied, also becoming a market sub-category, but the main literary movement has continued and developed into post cyberpunk movements.
Steampunk was originally a joking reference created by K.W. Jeter to refer to works that he, Tim Powers, James Blaylock and some others were doing in both SF and fantasy. These authors, though, again, formed a literary movement of their time -- the mid-late 1980's -- of similar stylistic and thematic approach that was parallel to but rather less punkish than cyberpunk, and as other authors entered into this area, especially for alternate history, it became a more definite movement based on using historical periods and interjecting anachronistic and futuristic technology into them to look at human interaction with technology and social upheaval (again parallel with cyberpunk.) It's the contrast of modern and old tech together stylistically -- which is different from cyberpunk -- that then caught on in fashion, jewelry and design. Punk became the go-to word for dividing stories up not into literary movements but into sub-categories based on the content of the tech. So we got not only dieselpunk for the World War era settings directly from steampunk, but other groupings of content like biopunk, splatterpunk, etc. Some of these sub-categories have stylistic and thematic aspects of their own, like splatterpunk, but others, like biopunk and dieselpunk, don't.
Dieselpunk is essentially steampunk pushed to a slightly later industrial age. So if you want to be very specific about the content of a book, you can say a novel set in WWII or I is dieselpunk as opposed to one set in the Victorian age. But as movements, dieselpunk is just a sub-set of steampunk. Likewise, a secondary world story can be steampunk if it sets up a medievalish world that nonetheless has industrial, technological or futuristic tech in that world, or if the imaginary world is similar in industrial tech to the Restoration, Victorian, early WWII, etc. but with odd, futuristic tech mixed in. You can also manage a dystopian post-apoc world, like Revolution, that has a mix of technologies, ranging from futuristic (the stuff that turned lights out,) to steam power in the loss of electricity. The sensibility of the story thematically is dealing with that mish mash of tech as a major focus of the story, etc., and including steam-era tech such as steam engines, airships, winding clockworks. Thematically, steampunk goes for swashbuckle and adventure more than punk rebellion of youth and noir style, although steampunk can be noir or horror and is concerned with major social shifts related to technology. So Leviathan, Sky Captain and the World of Tomorrow are generally considered part of the steampunk movement, even though they are not set in a steam-heavy era.
Steampunk and derivatives are also in some ways shielded in story world from any extinction by our love of time travel. Steampunk is mixing up tech of different time periods or alternate time periods or invented time periods (secondary worlds, post-apoc, etc.) and this is related to time travel in which tech can switch periods and many steampunk stories involve time travel in some way, and of course alternate dimensions. So steampunk continues to be popular as a stylistic and visual aesthetic in general culture and as a literary movement in story-telling with derivatives such as decopunk, cattle punk and diselpunk, well connected with time travel, multiverses and alternate history as plot/setting concepts. And in the last five years it has had an expansion because of the general expansion in fantasy including historical fantasy, renewed growth in SF where alternate history has remained strong, expansion for more areas in YA fiction, and an interest in industrial historical settings for movies and television. Right now they're trying to launch a steampunk t.v. series with an interactive Web component called Lantern City. So again, I'd put steampunk in the growing healthy, not dying on the vine level. It's also become a totally not unusual but very attractive design style, separate from the thematic ideas of the literary movement.