These are basically the same word in different forms, both meaning “dislike” or “hate.”
Loath is the adjective and loathe is the verb:
Adjective: I am loath to leave my mother when she is sick.
Verb: I loathe leaving my mother when she is sick.
Loath and loathe are polite words that you can use in place of hate. Because they are polite, they are not very emotional; for example, you probably would not shout “I loathe you!” when you are angry.