中文的概念大,既指中国汉语言又指中国汉文学；汉语的概念小,只是指汉语言. 因此,大学的中文系有中文专业,有汉语专业.中文专业汉语、文学兼学,而汉语专业则专攻汉语言. ----摘自网络

There are actually a lot of words for Chinese. This is my understanding. 中文 - Chinese. This means both spoken and written Chinese. 文 means written language, so like Mr Zhang said, if people want to know whether you can read Chinese, they will ask you 你會寫中文, not 你會寫漢語. 漢語 is the language 語 of the Han people 漢族. There are a a bunch of different ethnic minorities here in China 民族. The Han people are the biggest majority. That's where 漢語 comes from. 中國話 - The speech of China. Older people often say this. 普通話 - The language of the common people. This is used to refer to modern day Mandarin, and refers to speech only. This sounds a bit communist, so Taiwanese people will often tease you if you say it, and never say it themselves. 國語 - The national language - this is what they call Mandarin in Taiwan. 華語 - the language of 華人 (ethnically Chinese people). This is what Mandarin is called in Singapore.

你好，熊力！ 关于两个词的用法，“汉语”是说的，而“中文”能说能写。比如说，通常我们会问：“你会说中文吗？”、“你会说汉语吗？”或者“你会写中文吗？”，但是我们不怎么会问“你会写汉语吗”？当然也有人会这样讲，但是我很少听过“写汉语”；大家都说“写中文”。 希望我帮了你忙 =)

In modern times, for most people (both native speakers and foreign learners) , the only difference is that “中文” covers both spoken words and literature; while “汉语” only refers to as speaking. That's why you can write “中文” but not “汉语”. That's because “普通话 (Mandarin)” is compiled largely based on “汉语”, and has been set as the official language of China. Strictly speaking, there may be no precise definition of “中文”. It may refer to as including hundreds of dialects and ethnic languages (满语 Manchu language；藏语 Tibetan languages；蒙语 Mongolian language, etc.), which are/were used in different parts of China or historical periods. In this case, “汉语” is just a part of “中文”.

See my comment in my correction for https://www.italki.com/notebook/4206474/entry/854292