For its part Tokyo believes that all such issues were resolved through a treaty to normalise relations in 1965, which included massive economic aid to develop the South, at the time still recovering from the ravages of the Korean War.
Moon himself told Japan's Yomiuri Shimbun newspaper in an interview published Tuesday that he supported "future-oriented cooperation", separate from the issues of history.
But at the same time, he said that "true reconciliation" was not possible unless a "sincere self-reflection and an apology from the bottom of the heart must be conveyed to and received by the victims".
Analysts say the two countries should try to draw a line under the past in favour of "more diplomatic options".
"Korea and Japan both face a lot of shared challenges - North Korea, an unpredictable United States, an aggressive China, and the difficulty of sustaining economic growth," Mintaro Oba, a former US State Department official, told AFP.
"Cooperation between the two governments is both possible and critically important."
Daniel Pinkston, a lecturer in international relations at Troy University, added: "It is in the interest (of) both countries to resolve the issues."
Koreans suffered immensely under Japan's colonial rule, Pinkston said, but Seoul now had a shared responsibility to resolve the issue with "some maturity and strength".
"Japan today is not the Japan in the 1930s," he said. "That was then and this is now."
Japan has repeatedly addressed its wartime atrocities, notably the 1993 Kono Statement on the comfort women issue and a landmark apology by prime minister Tomiichi Murayama in 1995.
But remarks and actions by its own government and politicians have fuelled South Korean distrust of Japan's sincerity, notably the regular appearances by MPs at Tokyo's controversial Yasukuni Shrine, which honours millions of Japanese war dead but also senior convicted war criminals.
The nationalist Abe's own grandfather was arrested as a war crimes suspect but never charged, and a recent poll by Seoul's Asan Institute showed he was South Koreans' least liked leader.
"Tokyo's apologies have been perceived as too little, too late," writes Jennifer Lind of Dartmouth College in her book "Sorry States: Apologies in International Politics".
"Even worse, its politicians repeatedly shock survivors and the global community by denying past atrocities; its history textbooks whitewash its wartime crimes," Lind added.
South Korea and Japan bicker over statues of a comfort woman installed by activists in front of the Japanese embassy in Seoul and elsewhere.
Park Geun-hye, Moon's ousted predecessor, struck a deal in December 2015 with Tokyo under which Seoul promised not to raise the sex slaves issue again and Japan paid one billion yen (US$8.9 million) to a foundation dedicated to supporting the victims.
But Tokyo fell short of taking legal responsibility under the agreement, which angered some survivors.
Moon has not torn up the deal, but has called it a "wrongful" solution and said the South would return the money.
That "aftermath" has further complicated efforts to resolve the comfort women issue, said Ha Jong-moon, a professor of Japanese Studies at Hanshin University.
Seoul needed to find a deal with Tokyo that was acceptable to its own public, he said, but added: "The gap is too huge between what South Korea needs and what Japan is willing to do."