No 和 Not 都表示否认，两者主要区别：No 修饰名词，而 Not 修饰动词和形容词:
No man has ever walked on the Moon before Neil Armstrong.
在尼尔·阿姆斯特朗之前，没有人在月球上行走过。
They have not invited Lucy to the dinner.
他们没有邀请露西参加宴会
另一个区别是，Not 常和其他短语或从句一起使用：
It’s not often that you stop and think about the way you breathe.
你停下来思考自己的呼吸方式，这不是常见的。
Not suitable for children under 15.
不适合 15 岁以下的小朋友。
Not surprisingly, it was a tense match but eventually the more experienced Australians won.
不出所料，虽然是一场紧张的比赛但最终还是经验更加丰富的澳大利亚人获胜了。
A: Do you go cycling all year round? 你全年都骑自行车吗？
B: Not in the winter. 冬天不骑。
No 和 not any 有一些差别：
There’s no reason to be afraid of her. = There is not any reason to be afraid of her.
没有理由去怕她。
There are no eggs in the fridge. = There are not any eggs in the fridge.
冰箱里面没有鸡蛋。
no 比 not any 表达否定含义更加强烈：
| no | not any |
|---|---|
| He paid no attention to what I was saying. | He did not pay any attention to what I was saying. |
| I’ve decided that I have no alternative. | I’ve decided that I do not have any alternative. |
一个名词如果表达完整不可分割的对象的话，不能用 no，而要用 not a/an：
A potato is not a fruit.
A potato is no fruit.
这里 potato 就是一个完整不可分割的对象。
此外，可以用 no，等同于 not a/an：
It’s no secret that we are interested. = It’s not a secret.
去回答 yes-no 的问题，或者同意别人的否定表述，我们用 no 而不用 not：
A: Do you need anything from the shops? 你需要从商店买些东西吗？
B: No. I went earlier, thanks. 不用。我早些时候去过了，谢谢。
A: He’s not going to get any better. 他不会好起来的。
B: No. You’re right. 不会了，你是对的。
Not: Not. You’re right.
一些固定表达：no problem，no good，no chance，not worth
We use no and not in some common fixed expressions:
A: When you see Alan, can you give him this letter? 你看到艾伦的时候，可以把这封信交给他吗？
B: Yes sure, no problem. 可以，没问题。
She had no idea what time they were arriving. 她不知道他们什么时候到。
It’s no good standing around watching. Do something! 站着旁边围观不太好，做些什么！
You’ve got no chance of getting a ticket now. They’re all sold out. 你现在没机会得到票了，他们都卖光了。
It’s not worth taking a taxi. We can walk. 不值得打车，我们可以走路。
Reference
[1] Difference Between no and Not
[2] No or not?