It's all about co-location of how these words go. 'Merry' often goes with 'Christmas' and 'Happy' often goes with 'New Year'. If you switch them around they would not be wrong. It just wouldn't sound suitable. In fact in a movie called 'Coming to America,' Eddie Murphy, an African prince who is new to America, shouts 'Merry New Year!' This scene is funny as it sounds weird and hilarious.
Christmas is a present-tense event, existing for a relatively short period of time--24 hours. The new year is a future-tense on-going reality, which is not experienced all at once.
The term merry is used to describe emotions, or state-of-being only. Happy on the other hand can be used to describe emotions, a state-of-being, or a NOUN.
Thus when you wish someone a merry Christmas, you are wishing that their emotions or state-of-being be merry all day. The emphasis is on the state of being. (Usually because you are spending time with family, and experiences traditions etc).
Conversely, when you wish someone a Happy New Year, you are wishing that their YEAR be good. The emphasis is on the YEAR. (If the year's events are good, the person could then also experience happiness.)
happy
/ˈhæpi/
–adjective, -pi⋅er, -pi⋅est.
1. enjoying or showing or marked by joy or pleasure; "a happy smile"; "spent many happy days on the beach"; "a happy marriage"
merry
/ˈmɛri/
–adjective, mer⋅ri⋅er, mer⋅ri⋅est.
1. full of cheerfulness or gaiety; joyous in disposition or spirit: a merry little man.
