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. The large window permits light to enter the room. 大窗户能让光线进入屋内。
2. Permit me to set them out. 请允许我把他们指出来。
3. When you back to China you can go on to do your Residence permit. 当你再次回到中国，你可以继承你的居留许可办理的流程。
英英解释
n.
1. a legal document giving official permission to do something
2. the act of giving a formal (usually written) authorization
3. large game fish; found in waters of the West Indies v.
1. consent to, give permission
2. make it possible through a specific action or lack of action for something to happen
3. allow the presence of or allow (an activity) without opposing or prohibiting
