heart
Heart emphasizes the inner resolve or spirit that keeps someone going despite hardship or discouragement, often in fixed phrases like 'lose heart' or 'take heart'; it rarely suggests boldness toward danger.
heart强调支撑一个人在逆境或挫折中坚持下去的内在意志力，常见于「lose heart」（灰心、丧失勇气）、「take heart」（振作起来）等固定搭配；它很少表示面对危险时的大胆。
She never lost heart, even after failing the exam three times.
即使连续三次考试失败，她也从未灰心丧气。
nerve 本词
Nerve highlights the boldness needed to face a risky, scary, or socially daring situation, often with a hint of audacity; typical collocations are 'have the nerve to' and 'lose one's nerve'.
nerve强调面对危险、令人害怕或需要冒失的社交场合时所需要的胆量，常带有放肆、大胆的意味；典型搭配有「have the nerve to」（有胆量／厚脸皮做某事）和「lose one's nerve」（临阵退缩）。
He didn't have the nerve to ask his boss for a raise.
他没有胆量向老板提出加薪的要求。

Heart and nerve both can mean courage, but differ: heart is the inner strength to keep going despite hardship, as in 'lose heart' or 'take heart'. Nerve is the boldness to face something dangerous or scary, as in 'have the nerve to'.

这两个词在「勇气」这一意义上很接近，但侧重点和语气不同：heart强调支撑人克服逆境、坚持下去的内在意志力，nerve强调面对危险或令人紧张的场合时敢于行动的胆量，有时略带放肆之意。此处不比较heart表示「心脏」的义项，也不比较nerve表示「神经」的义项。