Register split: spoken vs written
没 and 无 both express negation or absence but live in different registers. 没 is the everyday spoken word: 没吃饭, 没问题, 没关系, 没事. 无 is the essay, news, legal, academic, and classical-style word: 无效 (invalid), 无知 (ignorant), 无限 (unlimited), 无能 (incompetent). In spoken Chinese, you rarely produce 无 alone; you read it constantly in signage and writing.
没 as the standard negation
没 (full form 没有) negates past completed actions and existence with 有. 我没去过北京 (I have never been to Beijing), 我没有时间 (I do not have time), 桌子上没有书 (there is no book on the table). The 有 can drop in speech: 我没钱 (I have no money) is a shortened 我没有钱. This is the everyday negation you use hundreds of times a day.
我没有看过这部电影,但听说很好看。
Wǒ méiyǒu kàn guo zhè bù diànyǐng, dàn tīngshuō hěn hǎokàn.
I have not seen this movie, but I hear it is good.
无 in compounds
无 almost never stands alone in modern Mandarin; it lives in compounds. 无聊 (boring, literally "without interest"), 无论 (no matter), 无所谓 (indifferent / does not matter), 无意 (unintentionally), 无数 (countless). These compounds are common in both speech and writing. The 无 here functions as a word-forming morpheme, not a standalone negator.
这个问题无论怎么解决都很难。
Zhè ge wèntí wúlùn zěnme jiějué dōu hěn nán.
No matter how we solve this problem, it is difficult.
无 in formal writing
In news, legal documents, contracts, and academic writing, 无 replaces 没 in many contexts. 无担保 (no guarantee), 无风险 (no risk), 无偿 (free of charge, unpaid), 无效 (invalid / null). A contract might say 本合同无效 (this contract is invalid), never 本合同没效. In casual speech this sounds artificially formal. When RJ is reading Chinese, 无 is everywhere; when speaking, 没 dominates.
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Fixed expressions and idioms
Many set phrases lock in 无: 无法 (unable to), 无穷 (infinite), 无比 (incomparable), 无缘无故 (without reason). Swapping 没 into these creates wrong or nonsense phrases. Treat these as vocabulary units, not negations of 有. Conversely, set phrases with 没 (没办法, no way; 没意思, boring; 没事, no big deal) do not swap to 无 without sounding strange.