Their appearance different is the sentence has 'A' whether or not, but I have heard both... What's their difference? or no difference?
A happy new year
This would be said in a sentence like "have a happy new year!". It treats "happy new year" as an object.
Happy New Year
This is just another saying that you tell someone on new years.
Both "have a happy new year" and "happy new year" are both acceptable things to say to someone on new years eve/day.