期许 is more formal and earnest than 希望 (xīwàng, 'to hope') — it implies a sense of trust and confidence in the person being expected to deliver. Common in commencement speeches, mentorship contexts, and personal essays.
register
Often used as a noun (as in 对…的期许, 'expectations of…'). The verb sense is mildly literary; 期待 (qīdài) is the everyday equivalent.
