While there is an overarching theme of how science and science fiction influence each other, there is no one specific way to approach it. Heinlein, Asimov, and C.S. Lewis all discuss several types of science fiction and how these relate to scientific knowledge, some more critically than others (Turning Points 3-28, 39-61, 119-131). Asmov's distinctions seem the most well-reasoned, so I will use his.
The first form Asimov discusses he labels “adventure science fiction” or “space opera” (Turning Points 39). In this form of science fiction, the technology is nothing more than a backdrop for an otherwise conventional story. One example would be the Star Wars film series. Star Wars features technological advances such as autonomous robots, cloning, lasers, and easy space travel. But none of these affect the story in more than a cosmetic way: the lasers are used identically to regular weapons, which don't exist in this world; the robots and clones are simply stand-ins for the faceless grunts found in cla**ic war stories like The Iliad or any John Wayne movie; and the space travel is simply a method of transport. None of the technologies are explained and the focus of the story is not on how these changes may affect society; they are givens. Were Star Wars to be set in the ancient world with normal weaponry and ships instead of spaceships, nothing about the story would change.
The looseness that space operas afford an author allows for focus on the story, be that plot or characters. Conventionally, this focus is primarily on the plot, which is part of the origin of the term space opera—a play on the melodrama and plot focus of soap operas. But in the context Asimov discusses, this focus on plot is not a requirement; simply a convention. The only requirement is that the story focus on the characters and has the technology as a backdrop, permitting the author to simply invent technologies that sound interesting, so long as they don't explicitly contradict scientific facts of the time.
Because of this freedom, authors often come up with more extravagant technologies, such as Star Wars' lightsabers. In turn, these more extravagant ideas are more attention grabbing and often serve as the initial inspiration to pursue science, as can be seen by the huge number of people Star Trek influenced (How William Shatner Changed the World). These scientists may move on from these initial motivations or they may remain dedicated. By implanting these ideas and drawing people to pursue a discipline of science, such as Dr. Kaku with Flash Gordon, space operas have no doubt had a great impact on the advancement of science, and received new technologies from unexpected outcomes for future stories in return (curiosity.com).
The second form Asimov discusses he deems “gadget fiction” (Turning Points 40). In this form, the technology is the story. The focus of the plot is creating the technology, with the conflict coming from the difficulties encountered in the process, and the resolution is the completion and recognition by the inventor's society of the technology. Asimov gives a hypothetical story about a man inventing an automobile in the late 1800s. The difficulty the man finds in getting an internal combustion engine to work properly and relatively safely is the conflict, and the resolution is completing the automobile and winning a race against a conventional horse carriage.
Gadget fiction is the antithesis of space opera, requiring an intimate knowledge of the current state of technology and science, and appeals to a different audience of readers and authors. These stories highlight the intricacy of the tasks needed to complete new technologies and how rewarding such pursuits can be in and of themselves.
Like space operas, gadget fiction has no doubt inspired any number of scientists, but not because of the wondrous technologies in the stories. Rather, these readers might be fascinated by the thought of even inventing new gadgets and through these stories learn about ways they could pursue such pa**ions. Additionally, these stories might inspire those already established in science in a different way: by giving them potential ideas for approaching their own research. And in turn, advances in science give these authors new techniques for developing the gadgets in their stories.
The final, and probably most recognized, form is “social science fiction” (Turning Points 40). In this form, the technology plays a role beyond the cosmetic, but is not the entire focus of the story. One such example would be Asimov's novel The Gods Themselves. This story features multiple major advances in technology, including inhabited colonies on the Moon and a source of seemingly infinite energy. While certain things, like the technology to reach the Moon and build colonies are givens, other things, like the differences between living on Earth and the Moon and the source of seemingly infinite energy, are addressed in depth. But the story does not focus on the technologies and principles; it focuses on how these things cause conflict between the characters.
Because this form requires the technology to be an integral part of the story, it necessitates an understanding of the workings of the technology and the potential implications. In this way it is similar to gadget fiction. But unlike gadget fiction, the technology is not the entire focus so the story doesn't have to go into as much detail about the mechanics behind it. In looser examples, like Vonnegut's Cat's Cradle, the technology can remain largely unexplained making it somewhat similar to space operas.
However, unlike space operas, the technology isn't simply a backdrop to the story, but an integral part. Because of this, the author must acknowledge that the technologies and scientific knowledge present in the story have an effect on the characters and explore this relationship between the characters and science in one form or another.
This relationship between science and people gives social science fiction a distinctly different relationship between it and science. Such a relationship might inspire scientists to pursue an idea not because it simply seems “neat” or “cool” or provide potential insight into how to approach a problem. Instead, it might inspire them to pursue a technology that, while not flashy, might be incredibly useful or life-changing when applied to some aspect of the current, or even future, world. And by observing how scientific advances affect the world, authors may gain new insight into how the technologies they invent might affect the people that inhabit their worlds, potentially in ways they never expected.
All of these forms of science fiction are still fully in line with the relationship discussed, but each has a different goal and a different form of the relationship. All three can serve as inspiration to scientific advances and the creation of new technology, no matter how fanciful so long as it isn't fantastic (I'm still hoping for my own lightsaber), but their use of scientific knowledge and forms of inspiration are what separate them.