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In China, the dual-tracked celebration was originally made law by Yuan Shikai on 21 January, Third Year of the Republic (1914).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.