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It's true that in theory it would be easier to reach the moon with modern aerospace technolgies than with those of the 1960s, but no Space Agency possesses a launch vehicle usable for such a mission at present. The gargantuan Saturn-V rocket remains by far the most powerful spacecraft ever created. The Falcon Heavy, first launched last year, can deliver over twice as much material into Earth Orbit as the Delta IV, the next most powerful launch system in current use worldwide. Yet as strong as the Heavy is, it can carry less than half as much material into Earth Orbit as the Saturn V. So far, super-heavy rockets have not seen any use since the end of the Apollo program (though the USA and China have new super-heavy flights planned in the near future), and Apollo 16 remains the heaviest space mission in human history. Owing to advancements in computing and robotic technologies (as well as much lower payload weight), unmanned missions can now be carried out at far lower cost, while still returning impressive amounts of scientific data comparable to what may be obtained with a manned mission. This is why, especially since the end of the Space Shuttle Program in 2011, NASA has given the bulk of its attention to robotic Deep Space missions.
While Apollo's scientific achievements are extremely impressive and have been of great value to astronomers and geologists, it was incredibly expensive compared to other government-funded research. At its peak in 1966, NASA represented 4.41% of all Federal spending, an absolutely outrageous price. For comparison 4.4% of the 2018 US Budget is enough to cover funding for the Department of Education, Department of Health and Human Services, Department of Agriculture, Department of Energy, and the Small Business Administration, COMBINED. Though spaceflight as a whole has become critical to modern Global Telecommunications, most of these technologies are not especially dependent on Apollo Program advancements. Concerns that Manned Space Exploration "just isn't worth it" thus factor heavily into the decision making of Space Agencies today.
And remember--the United States was the world's largest economy by far in 1965, with over double the GDP of the Soviet Union and more than seven times that of Japan, the UK, or West Germany. Even today, following over half a century of the fastest economic growth in global history, only seven countries can claim a Nominal GDP higher than that of 1965 America. The Soviet Manned Moon Program is much more poorly known than Apollo, having been a state secret until 1990. It had to operate on a much tighter budget than Apollo and was rife with administrative problems.
It was a miserable failure. Not only did the Soviets fail to land men on the moon, as America first accomplished with Apollo 11, they could not successfully engineer a launch system that could be used for such a mission--something America had accomplished with Apollo 4 in 1967. While NASA celebrated its first successful Manned mission to orbit the moon, the Soviet N1--a super-heavy roughly equal in strength to the Saturn V--crashed to the ground within minutes of launch. While Armstrong, Aldrin, and Collins, prepared for the launch of Apollo 11, a second N2 launch exploded after hitting the launchpad 23 seconds into flight. By the time Cernan and Schmitt returned to Earth following Apollo 17, another 2 tests of the N1 had failed. The Soviet Union would not test the N1, or any other super-heavy rocket, again.
The Apollo Program, being a point of national pride and American scientific achievement, has aged very well in the public memory. But despite its modern reputation, Apollo faced significant criticism primarily due to the perceived excessive cost-to-benefit ratio. Consider the following timeline of the Apollo Program and opinion polling regarding it.
1960: Apollo Program Announced
1961: President Kennedy's Famous Speech, First American in Space, 42% in Favor of Apollo Program, 46% Opposed (June)
1962: First US Orbital Manned Flight.
1963: Project Mercury Ends.
1964: Project Gemini begins.
1965: First Gemini Manned Flight. 40% in Favor, 51% Opposed (February) 45% in Favor, 42% Opposed (October)
1966: Project Gemini Ends.
1967: Apollo 1 Disaster, First Unmanned Saturn V Launch. 34% in Favor, 54% Opposed (July)
1968: First Manned Apollo Missions. Apollo 7, 8 (launch)
1969: First Men on the Moon; Apollo 8 (reentry), 9, 10, 11, 12
1970: Apollo 13 Fails, but Crew Survives, 40% in Favor, 56% Opposed
1971: Apollo 14, 15
1972: Apollo 16, 17
1973: Apollo Program officially ends.
This is all to say that Apollo was a highly controversial and generally unpopular project at the time, the most expensive government-funded scientific project in human history, and conferred rather limited scientific gains. NASA's budget was continually cut between 1967 and 1974, forcing a premature end to the Apollo program (another 3 planned Moon landings had to be canceled), and ending Human Space Exploration beyond Earth's orbit up to the present day.
But the main reason that Apollo was funded in the first place, the Public Relations arms race against the Soviet Union, had melted away in the late 1960s. What successes the Soviets did have in this time were much more limited than those of the United States. Concern that the USSR could develop superior missile technology to the United States military was essentially dead and had already been so for years, and NASA's crewed spacecraft designs were military useless anyway. Following the end of Apollo, the US public remained unenthusiastic about NASA--56% of adults polled in 1975 said that they supported even further budget cuts to the agency. Between the massive economic investment necessary for a Manned Moon Landing Program, the lack of political or military pressure to develop manned spaceflight technology, and limited scientific benefits such a program would have over alternative research investments, manned travel to the moon remains unviable in the eyes of most Space Agencies, and is a distant goal at best for even the largest space programs.
NASA has focused chiefly on the development of more affordable orbital launch systems since the end of Apollo, most famously (and expensively) through the Space Shuttle Program (1977-2011), which intended to lower orbital launch costs by making spacecraft partially reusable. Difficulties with Shuttle launches, especially the Challenger Disaster (1986) led to NASA shifting its focus back to developing more traditional ELV launch systems, for both military (mostly reconnaissance) and civilian (mostly telecommunications) use. This remains one of the two primary focuses of NASA today, the other being the development of robotic spacecraft to explore far beyond Earth itself.