Quite a few things!
https://www.nasa.gov/50th/50th_magazine/benefits.html
1978: Teflon-coated fiberglass developed in the 1970s as a new fabric for astronaut spacesuits has been used as a permanent roofing material for buildings and stadiums worldwide. (By the way, contrary to urban myth, NASA did not invent Teflon.)
1982: Astronauts working on the lunar surface wore liquid-cooled garments under their space suits to protect them from temperatures approaching 250 degrees Fahrenheit. These garments, further developed and refined by NASA’s Johnson Space Center, are among the agency’s most widely used spinoffs, with adaptations for portable cooling systems for treatment of medical ailments such as burning limb syndrome, multiple sclerosis, spinal injuries and sports injuries.
1986: A joint National Bureau of Standards/NASA project directed at the Johnson Space Center resulted in a lightweight breathing system for firefighters. Now widely used in breathing apparatuses, the NASA technology is credited with significant reductions in inhalation injuries to the people who protect us.
1991: Tapping three separate NASA-developed technologies in the design and testing of its school bus chassis, a Chicago-based company was able to create a safer, more reliable, advanced chassis, which now has a large market share for this form of transportation.
1994: Relying on technologies created for servicing spacecraft, a Santa Barbara-based company developed a mechanical arm that allows surgeons to operate three instruments simultaneously, while performing laparoscopic surgery. In 2001, the first complete robotic surgical operation proved successful, when a team of doctors in New York removed the gallbladder of a woman in France using the Computer Motion equipment.
1995: Dr. Michael DeBakey of the Baylor College of Medicine teamed up with Johnson Space Center engineer David Saucier to develop an artificial heart pump – based on the design of NASA’s space shuttle main engine fuel pumps – that supplements the heart’s pumping capacity in the left ventricle. Later, a team at Ames Research Center modeled the blood flow, and improved the design to avoid harm to blood cells. The DeBakey Left Ventricular Assist Device (LVAD) can maintain the heart in a stable condition in patients requiring a transplant until a donor is found, which can range from one month to a year. Sometimes, permanent implantation of the LVAD can negate the need for a transplant. Bernard Rosenbaum, a Johnson Space Center propulsion engineer who worked with the DeBakey-Saucier group said, “I came to NASA in the early 1960s as we worked to land men on the moon, and I never dreamed I would also become part of an effort that could help people’s lives. We were energized and excited to do whatever it took to make it work.”
Errr... This is not how things usually are...
When moon mission was thought about, there were already existing Missiles which could travel to moon. So science was all in place. Technology was partially done. Remaining part of technology was needed.
Now coming to curing cancer. EVEN TODAY! we don't have enough science for cancer. YES! EVEN TODAY! That means we are FAR from curing cancer. We cann't develop technology be cause we don't know what is going on in cancer. Plus Cancer is not ONE disease. It is a group of 100 diseases. And we don't know enough about even one of them.
So to put it in perspective, you have
70% chance of reaching moon in 1960
less than 1% chance of "curing cancer" (I don't know what it even means) in 1990, let along 1960.