There was an era of celebrity-physicists, including names such as Einstein, Feynman, Schrodinger, Oppenheimer, etc. in the first half of the 20th century - I'm willing to bet that if you're not a physicist, you'll likely recognize most of those names. On the other hand, the prominent mathematicians of that era, or biologists of that era, etc. did not achieve the same celebrity status. Stephen Hawking was something of an anachronism in his lifetime, of a prominent physicist who was also a celebrity, and there hasn't been a figure with the same level of prestige since then.
What was it that allowed early 20th century physicists to become celebritized? Why did this trend seemingly go away?
The number who had celebrity prior to 1945 were pretty slim — Einstein was the most famous (because of the vindication of the theory of relativity in Eddington's 1918 experiment, which made front page newspaper globally), and there were a few others whose discoveries were notable enough that they would appear in science textbooks globally (like Bohr and Fermi). But they didn't achieve superstardom until after World War II, when the atomic bomb was credited for ending the war and changing the world, and theoretical physicists in particular were attributed an almost mystical connection to its success. The amount of credit they got for it was fairly out of proportion to the importance of theoretical physics to build said weapons, in part because the theoretical physics was the easiest to declassify (because it was mostly based on pre-war discoveries); the chemists, metallurgists, and engineers on the project got essentially ignored, because their work was still classified as it had direct relevance to actually making these weapons. With such fame came heightened scrutiny as well; David Kaiser has shown that theoretical physicists got targeted more often than any other academic discipline during McCarthyism — to be powerful is to also potentially be dangerous.
The mid-20th century was sort of a Golden Age for theoretical physics, as well, buoyed by the success of several major new theoretical breakthroughs (quantum chromodynamics, quarks, the Standard Model), themselves verified and encouraged by the massive expenditures on basic physics that took place in the mid-Cold War, itself a product of a social belief that theoretical physicists were necessary to win the Cold War.
The decline in their prominence is both a decline of that particular generation of physicists — they all eventually died — and the fact that support for basic science waned after 1970. After filling in the Standard Model, the theoretical breakthroughs have been relatively modest — the field is largely dominated by Theories of Everything that as of yet cannot be tested (e.g. string theory and its variants). There are still some physicists who have made plenty good on popularizing this work (e.g. Brian Greene) but even if it is true it is not clear that these theories get us anything, and it's not clear they're true (we have no way to tell). The physicist Steven Weinberg described the state of things (in the mid-1990s) as such: "There’s never been a time when there’s been so little excitement in the sense of experiments suggesting really new ideas or theories being able to make new and qualitatively different kinds of predictions that are then borne out by experiments." (This quote comes from John Horgan, The End of Science, which is a pretty provocative text.)
Hawking is interesting because his celebrity was also clearly linked to his health; A Brief History of Time is a best-seller that practically nobody read. Having a genius brain incased in a crippled body makes for good cultural resonance — the ultimate expression of Carteseian dualism, and a good metaphor for how most everybody feels as their bodies age.
Anyway. The short answer is, "the bomb," but as you can see there are other connections there. It is also worth noting that only a handful became true celebrities. Everyone knows Feynman because of his "funny" books and his appearance at the Challenger post-mortem. Far fewer people know Murray Gell-Mann, whose work arguably eclipses Feynman's. Lots of people know Oppenheimer, who has been memorialized in plays, novels, television shows, films, and even an opera. Very few could name the only person to win two Nobel Prizes in Physics. The answer is >!John Bardeen — for the transistor and superconductivity.!< As one of his biographers (Lillian Hoddeson) has noted, he was not eccentric, or funny, or exciting. He doesn't make for a good story.
Some reading of interest:
On the Golden Age of physics, see Daniel Kevles, The Physicists (1978)
On the rise of physicists after the bomb, and their targeting during McCarthyism, see David Kaiser, "The Atomic Secret in Red Hands? American Suspicions of Theoretical Physicists During the Early Cold War," Representations 90 (Spring 2005), 28-60.
On Hawking, see Helene Milalet, Hawking Incorporated (2012)