Difference between 汉语 and 华文
It should be 华语. May be it is a Singapore thing to use the term 华文. 

华 or 华夏 is the origin of Chinese culture. While 汉 is one of the greatest and longest ruling dynasty in Chinese history. The words 华 and 汉 along with 唐 (another great dynasty) are commonly used as representative of "Chinese"


华语, 汉语 and 唐話 (Cantonese) all mean "Chinese spoken language" (中国语)
Both 汉文/ 汉字 and 唐字 (Cantonese) mean "Chinese written language" (中文/中国字)
Both 华人 and 唐人(Cantonese) mean "Chinese people" (中国人)