Why is it “Merry” Christmas, but “Happy” New Year?

Solution 1:

Looking at the definitions of merry and happy, I would suggest that merry implies more short-term jollity, and happy more long-term contentment. That would square well with the Merry Christmas and Happy New Year usage.

The definitions of merry I found here and here, and for happy here and here.

It might also be worth noting that merry is used in phrases like play merry hell, play Merry Andrew and the alternative meaning of merry (meaning 4) meaning somewhat inebriated, which are all short-term (one hopes) states of affairs.

Solution 2:

mer·ri·ment
a state of enjoyable exuberance; playful fun
noun /ˈmerēmənt/  Gaiety and fun
- her eyes sparkled with merriment

Happiness
Hap"pi*ness, n. [From Happy.]
1. Good luck; good fortune; prosperity.
- All happiness bechance to thee in Milan! Shak.

Merriment is having some good fun, while happiness is a much broader term, relating to overall well-being with all necessities of life, and so on.

Christmas is a time of celebration and therefore a 'Merry Christmas' would be appropriate. A new year, as in 'Happy New Year', on other hand, extends over a whole year (and further on) and as such the sense of good luck, good fortune and prosperity provided by Happy would be appropriate.