|Clearly, Old New Year is still a big deal!|
Today, 14 January,* is the "original" New Year's Day in the Julian calendar at the time that the Gregorian calendar was devised to replace it. (This switch was discussed in several places, most notably here and here.) As can be expected, not everyone adopted the new calendar system right away. Some hung onto the Julian calendar for a long time; in fact, it wasn't until 24 January 1918 that Lenin made the Gregorian calendar the law of the land in Russia.
By the early 20th century, the Julian and Gregorian calendars were separated by two weeks (not the 7 days that were the case centuries earlier in the time of Gervase of Canterbury, or even the 10 days that were the case in the 16th century when a whole chunk of October disappeared from European countries).
It turns out that Lenin did not actually change New Year's Day in Russia so much as he created a second one. In Russia it is the Old New Year, a day of nostalgia and feasting. In the Eastern Orthodox liturgy, it is the Leave-taking of the Christmas season: the final "wrap-up" of Christmas.
The Old New Year is still remembered in Serbia, Georgia, Bosnia-Herzegovina and Ukraine. It is celebrated in Wales as Hen Galan ["old new year"] (at least in Gwaun Valley in Pembrokeshire) when children receive sweet treats and sing carols. Likewise, some parts of Switzerland keep up the old tradition of Alter Sylvester, described and illustrated here.
Thanks to the refusal of protestant countries to change their dating systems based on a Roman Catholic idea, some very old customs have survived into modern times.
*Sometimes Old New Year is the 13th.