允许 (yǔnxǔ) is the standard, neutral verb for 'to permit, to allow' in both speech and writing. It is more formal than 让 (ràng, HSK2), which is the everyday spoken word for 'let/allow' and can also mean 'make (someone do something)' or serve as a passive marker. 许可 (xǔkě, HSK6) is a formal noun/verb used in official contexts like permits.