华语 (Huáyǔ) vs. 普通话 (Pǔtōnghuà): This is the most important distinction. 普通话 (Pǔtōnghuà) means “common speech” and is the official, standardized Mandarin language of the People's Republic of China (PRC). It's a political and educational standard. 华语, on the other hand, is the preferred term in many overseas Chinese communities (like Singapore and Malaysia) to refer to Mandarin. It's less a political standard and more a marker of cultural identity. A simple analogy: think of the difference between “The Queen's English” (a specific standard) and “English” (the global language spoken in many accents and variations). 普通话 is like the former, while 华语 is philosophically closer to the latter.
华语 (Huáyǔ) vs. 汉语 (Hànyǔ): 汉语 (Hànyǔ) means “the language of the Han people,” referring to China's largest ethnic group. This is the most common term used in academic settings and language textbooks, both inside and outside of China. It's a neutral, linguistic term. 华语 is more emotional and cultural, often used in media, music, and community contexts to foster a sense of belonging among the global Chinese diaspora (`华人`, Huárén).
A Global Identity: The use of 华语 highlights the value of a shared cultural heritage that transcends national borders. The global “C-Pop” or Mandopop music scene is called the 华语乐坛 (Huáyǔ yuètán)—the “Huayu music world”—because it includes artists and fans from mainland China, Taiwan, Hong Kong, Singapore, Malaysia, and beyond. They are all united by this common language, 华语.