Before the Meiji Restoration, Japan used the Chinese calendar and celebrated Chinese New Year just as China did.
Nowadays, both Japan and China have adopted the Western calendar. However, Japan celebrates Western New Year while China still celebrates Chinese New Year.
Why does today's Japan celebrate Western New Year instead of Chinese New Year? Also, why China still celebrate Chinese New Year despite having adopted Western calendar?
The short answer is because the Japanese government does not designate the old lunisolar new year as a public holiday.
Officially, China does in fact celebrate New Year's Day (元旦) on the Western (Gregorian) 1 January. In contrast, the traditional lunar new year is a public holiday named Spring Festival (春节). Since the latter is a longer holiday, combined with ancient traditions, it predominates in people's minds as the real new year's day. This is not, by the way, limited to China; Vietnam, Taiwan and Korea celebrate the event in a similar fashion.
In China, the dual-tracked celebration was originally made law by Yuan Shikai on 21 January, Third Year of the Republic (1914). The other major days of Chinese traditions, the Dragon Boat Festival, the Mid-Autumn Festival, and the Winter Solstice were declared public holidays at the same time (as Summer, Autumn, Winter festivals, respectively). Thus, while the Gregorian calendar was adopted to bring China in line with the West in civil administration, the old lunisolar calendar retained its cultural significance. The preservation of customary holidays had basically been an integral state policy from the start. Later attempts to change this met with popular opposition.
When Japan adopted the Gregorian solar calendar in 1873, the country was pursuing westernisation very comprehensively. The belief among elites at the time was that Asian practices were inferior to Western ones, and ought to be replaced by the latter (脱亜入欧). Thus the calendar was adopted as a full replacement, with no provisions made to perpetuate a dual-tracked celebration system like other Asian states deployed. When the old lunar calendar was phased out completely in 1910, it also largely died out from daily life.
Note that some temples and shrines in Japan continue to commemorate the Old First Month, 旧正月, and the general public is usually aware of the date.
It is not an either/or situation. In China they celebrate Gregorian New Year on 1 January and the traditional lunar New Year as well. It is the same in Vietnam.
It's a difference of emphasis and philosophy. Both Japan and China have officially incorporated the western New Year into their calendars.
The Chinese philosophy regarding such matters, however, was "Zhong xue wei ti, xi xue wei yong." (Chinese learning for the core, western learning for use.") Hence, China pays "lip service" to western practices, and goes its own way.
On the other hand, the Japanese ethos at the time of the Meiji emperor (the time the decision was made) was that West is best.
Your question seems Eurocentric. Why would a culture want to give up its traditions and replace it with Western traditions? Many non-Western cultures around the world may have adopted Western dress (suits, tie, etc.) but still retain their traditional clothing, such as India. Why do Indians still wear traditional clothing when they have Western clothing available? I believe it's because Indians are proud of their history, culture, and heritage and see no reason to replace their traditional clothing with Western-style clothes.
Are you aware of the history and meaning of the Spring Festival (Chinese Lunar New Year)? It goes back thousands of years to a time when Europe was just full of small kingdoms squabbling over power. In those days China was already a huge empire actively trading with the empires of the Middle East. Why would the Chinese want to give up the Chinese Lunar New Year--an ancient and beautiful holiday designed to help us reconnect with family members and celebrate the new year? Why would we want to replace our ancient traditions with the traditions of the West that have no meaning for us?
