I have asked this question to many Japanese people, and nobody was able to answer me. I have searched the Net, and finally found on Wikipedia that the Japanese New Year used to be the same as the Chinese/Korean/Vietnamese one until 1873.

In the Julian calendar (used in Europe from 45BC to AD1582 or 1752 or 1923) the official New Year was 1st March (Roman date) or 25th March (Christian date), and not 1 January (although it was already celebrated on that day too). The Venetian for example kept celebrating the Roman New Year on 1 March until 1797, when their republic was dissolved by Napoleon.

Nowadays, the New Year is still celebrated at different dates around the world.

It is interesting though that the Japanese should now celebrate New Year on 1 January like Westerners, and send greeting cards like Westerners too. I also wonder why almost all the Japanese I have met had no idea that the traditional Japanese New Year was the same as the Chinese New Year, although everybody seems to know that the Japanese Zodiac is (about) the same as the Chinese one.