Which is the correct term?

Which is the correct term to use when saying Happy New Year? Xīnnián kuàilè! or Gon Xi Fa Cai? 

新年快乐 (xīn nián kuài lè) definitely means "Happy New Year".  I have been exchanging that greeting with some of my Chinese friends.

I believe 恭喜发财 (gōng xǐ fā cái) means "Happiness and prosperity", something you say when you are giving and receiving gifts, e.g. 红包 (hóng bāo).
 

Thanks Robert. I was at a bakery in the Asian Market, and actually had the courage to speak some Chinese to her, and asked her about it (in English) and that is what she said also.

Today, in Taiwan, most people are greeting each other on the street with xīn nián kuài lè; however, some people simply say gōng xǐ.  The intent is the same in both cases.

Language and culture are  living so they change with time. What I remembered might have been different today.

I was in Hong Kong long long time ago during Chinese New Year. People just say gōng xǐ fā cái (the writing is the same but the sound is entirely different from Cantonese) in lieu of everyday greetings for the 15 days of the new year celebration in particular. Keep in mind Hong Kong Chinese is different from Taiwan Chinese and Mainland Chinese due to geopolitical differences.

The reason behind the saying gōng xǐ fā cái was that wishing adults you knew very well to barely register a year of prosperity was very important; Chinese is pragmatic. For children they said something else like wishing them excelling in their education endeavor. 

Adults, single and especially married, DO NOT receive hóng bāo except from their parents if they are still alive, only children so they rarely say gōng xǐ fā cái because prosperity and wealth meant nothing to them. 谢谢 xièxiè  from Mandarin speaking children 多谢duōxiè said in Cantonese from Cantonese speaking children, no different from English speaking children when they received a present.  多duō means more so it is the English equivalent of "thank you very much".  Single unmarried adults, regardless of age do not give hóng bāo to children. 

All married women carried a big pocketbook or purse and had 3 bundles of hóng bāo of different values. To those children they barely knew, like neighbors' they gave the smallest amount, those of their friends' children,  moderate and their unmarried nieces and nephews the most. It always came in pairs, one from the husband and the other from the wife (that's why the women carried the big purse; men didn't carry man-purse and pockets were not large enough) unless one was widowed then they gave one only. 

I hope this helped.