allow 普通用词，侧重听任、默许或不加阻止。在正式场合可用来表客气的请求。
let 常用词，用于各种非正式场合，语气最弱，指允许或无力阻止某事，暗示漠不关心或听之任之。
permit 正式用词，在多数场合可与allow换用，语义最强，指准许某人做某事，含权威或正式的意味。
leave 侧重不加干涉。
authorize 语气最强，指权威性的允许与认可。
参考例句
1. They allowed the children to go to the park. 他们让孩子们到公园去。
2. Allow me to introduce my boss for you! 请允许我介绍我的老板给你认识。
3. The facts allow of no other explanation. 这些事实不允许有其他解释。
英英解释
v.
1. make it possible through a specific action or lack of action for something to happen
2. consent to, give permission
3. let have
4. give or assign a resource to a particular person or cause
5. make a possibility or provide opportunity for; permit to be attainable or cause to remain
6. allow or plan for a certain possibility; concede the truth or validity of something
7. afford possibility
8. allow the other (baseball) team to score
9. grant as a discount or in exchange
10. allow the presence of or allow (an activity) without opposing or prohibiting
