From the limited understanding I have, both countries tried to reform their military to Western standards during the late 19th century. However, the Japanese reforms seemed to succeed better as they were able to defeat the Chinese in the first Sino-Japanese war despite numerical inferiority. Later, in the 20th century, the IJN grew into a force that could threaten the US Navy while the Chinese navy remained relatively small. The Japanese army remained qualitatively superior to chinese ground forces as well for most of the war. What were the reasons for China's military ineptitude?
This is an age old topic.
My answer is pretty simple. Japan could modernize because it became a nation-state by using ethnic nationalism. Qing China (1644-1911) could not modernize because it was multiethnic empire ruled by the minority Manchus.
So in Japan's case, the new Meiji state (1868-1912) could rely upon the age-old imperial family (going back to at least the 500sAD) and a relatively quick construction of Japanese identity based upon a shared language and culture. So despite the massive changes brought upon by Westernization, they could say that they remained Japanese and continue to swear loyalty to the new Westernizing government.
In Qing China's case the imperial family was Manchu. These Manchus were a non-Chinese people who came from north of the Great Wall and conquered the Han Chinese in the 1600s. The Manchus spoke and wrote in their own language, they were Tibetan Buddhists, and culturally they were semi-nomadic. The Chinese wrote in classic Chinese, they were a fusion of Daoism, Buddhism, and Confucianism, and they were agriculturalists who practiced foot-binding. To govern this China (that also included other minorities like the Mongols, Tibetans, and central Asian Muslims) the Manchus set up a system of diarchy where top government posts in China proper were divided evenly between Manchus and Han Chinese.
The only thing that really bound the Manchus together with the Chinese was Confucianism. The Manchus adopted Confucianism as the philosophy to govern China and continued to use it as the basis for the Civil Service Exams that selected government officials.
Fast forward to 19th century and the need to reform. One of the things that both countries needed to do was learn Western technology and adopt schools that taught many of the Western sciences.
In Japan, adopting that Western curriculum to modernize its military did not endanger Japanese identity nor endanger the basis of Meiji government. (Because of the ethnic nationalism mentioned above.)
In Qing China, however, if you adopted the Western curriculum you were abandoning the Confucian curriculum and the Civil Service Exams. It took a bright person at least 20 years of studying classical Chinese and Confucianism to pass the Civil Service Exams. It was unrealistic to add another 10 years of Western education. So if you pull out Confucianism from the curriculum, then what binds the Manchu people with the Han Chinese people? Very little I'm afraid. So when members of Qing China's leadership realize this, they pull back from full scale Westernization and reforms. Chinese students who study a Western curriculum instead of the Confucian curriculum also begin to question why they have to have an alien Manchu emperor in China. Thus it wasn't accidental that the father of the 1911 Revolution in China, Sun Yat-sen, got his education in the West (Hawaii and Hong Kong). And that the 1911 Revolution starts right after the Manchus abandon diarchy and staff 80% of the top government posts with Manchu princes in 1910.
TL:DR Japan could push through the necessary reforms to learn how to use ships and guns they bought, but also to eventually make them. Qing China could only buy ships and guns but not teach the students how to make them.
This also explains why once China became ruled by the Han Chinese, they could do lots of reforms (particularly social reforms) in the 20th century.