A demagogue (pronounced \?de-m?-?gäg\ or dem ah gog) is a leader, usually political, who makes false promises and gains his or her popularity through feelings rather than logic. It may also be spelled demagog. Though the vast preference is for the longer spelling. So much so that we could not find an acceptable example of the shorter spelling.

The actions of a demagogue can be termed demagoguery or demagogy, both of which have multiple accepted pronunciations. It can also be a verb for a person who acts like a demagogue.

A demigod can be one of two things. Either it is a person who is the child of a human and a god, mostly found in the Greek and Roman mythologies. In modern days, a demigod can also be someone who is so powerful or awe-inspiring that they seem to be like a god. It is always one word without a hyphen.

http://grammarist.com/spelling/demagogue-vs-demigod/