I sometimes see these two words mixed up in writing, and I sometimes hear them mixed up in speech.

Problem:
“Loath” and “loathe” are not synonyms.

Explanation:
“Loath” is an adjective that means reluctant or unwilling. “Loath” is usually followed by “to”, as in “Mary was loath to call her sister about their brother’s death.”

“Loathe” is a verb that means abhor, or dislike greatly, as in “Mary loathes making cold sales calls.”

I sometimes hear the verb “loathe” where the adjective “loath” is required. Misuse of “loathe” as a substitute for “loath” — especially in speech — seems to support my “Devolution toward Simpler” hypothesis. The verb “loathe” is simpler to pronounce (with the “th” sound like in “smooth”) than is the adjective “loath” (with the “th” sound like in “thin”).

Solution:
Use “loath” as an adjective (usually followed by “to”); use “loathe” as a verb. Remember that “loath” and “thin” have the same “th” sound, whereas “loathe” and “smooth” have the same “th” sound.