They are but the latter term is a bit more flexible. A mistress, in this context, is a woman who has a sexual relationship with a person who is married. A lover can be a man or a woman who has a sexual relationship with another person who may or may not be married or in a committed relationship.

Just to complicate the topic, a mistress can also be a woman that is dominant, for example a dominatrix.

Brianna is quite right. I would add that the word "mistress" is starting to become an archaic word that we don't use much. It is useful in a society in which powerful men "take" mistresses. There's no equivalent word for a powerful woman who takes on a male lover. To refer to a woman as a "mistress" carries a suggestion of a "kept woman," that is, a woman in a subordinate position living outside of society's norms. In today's society, you're usually better off using the word "lover."

No, and it could get you in trouble if you use one when you mean the other. A mistress is someone that a person has an affair with when they are dating someone else or when they are married to someone else. A lover is a romantic/sexual partner. A lover could be someone's wife, husband, girlfriend, boyfriend, someone they are having an affair with, or someone they just have sex with. A wife or a girlfriend is a lover, but not a mistress. A mistress is a mistress and a lover.