What is the difference of the meaning of 挂牌 in 挂牌督办 and 挂牌上市?
There won't be a real sign hanged.

挂牌 originally means setting up a sign as you know. When use it in 挂牌督办, since only authority department can 督办, 挂牌 means the department show the case to public to urge other lower department which is in charge of the case. 

The case under 挂牌督办 always has a deadline, so the lower department have to get it done in time, otherwise somebody would get things real bad. 

As 挂牌上市, that is a totally different commercial definition, it means a company list their shares in stock market.