Pleco defines them as:
认可 - approve
ex. phrase: 得到领导的认可
许可 - permit; allow:
ex. phrase: 凡是条件许可的地方
AND
得到领导的许可
So, according to this it seems both are pretty much interchangeable, at least in some situations. And last we have:
允许 - permit; allow
With the same definition as 许可. Can someone help me make it clear what the differences are with these three verbs are. Would be much appreciated!
认可accept, recognize, approve
观众并不认可他的演技。
The audience has not found his acting acceptable.
该专利申请已被专利局认可。
The patent application has been approved by the Patent Office.
许可 approve, permit; approval, permission (usually it is someone in authority that 许可s somebody to do something )
未经作者书面许可，不可转载本文。
This article may not be reproduced without written permission from the author.
未经许可，不可将图书带出阅览室。
The book cannot be taken out of the reading room without permission.
得到父亲的许可
Permitted by one's father
许可证 license
允许 allow
没我允许，周六周日都不许看电视。
You cannot watch TV on weekends unless you are allowed. (Parent speaking to son/daughter)
谁允许你那么做的？
Who gave you the right to do that? (Could be someone in authority speaking or a guy like you and me.)