#in shi'a islam. mutah is a temporary marriage

[tamamita]

Is someone who is "permanently" married allowed to enter a mu'tah marriage to another person, without consent or knowledge of their "permanent" spouse? (I use the word permanent because im unsure of the word for the more permanent marriage)

No, explicit consent is needed

[tamamita]

What is the difference between a Shia wedding and a Sunni wedding?

Shi'a don't require witnesses when the marriage contract is being signed. However, it is recommended to have at least two witnesses present. It is mandatory to have two witnesses when a divorce takes place. Sunnis require you to have two witnesses during the marriage ceremony, while it is not required to have witnesses during a divorce ceremony.

There is the concept of temporary marriage (Zawaj al-Mutah) within Shi'a Islam that allows partners to engage in a relationship for a fixed time. The purpose of such marriage is sexual gratification. Although the concept exists within the fiqh of Sunni Islam, it has ever since been banned by Umar.

Shi'as prohibit permanent marriages to the People of the Book (Jewish people, Christians, Zoroastrians, Sabeans and various Abrahamic religions) but allow for Shi'as to be temporarily married to them; Sunnis allow it in permanent marriage. Shi'as are allowed to be in a permanent marriage with Muslims of different branches of Islam.