A teacher is someone who provides knowledge, lessons, rules and information. Teachers are people who influence you in ways that can only be obtained from knowledge, books and memorization and the like.
A mentor is someone who provides those life lessons from experience, guides you with each step and sees you through no matter what. For example, my husband was a pilot. He was also a flight instructor. He sat with pilots, gave them flight instruction, classroom instruction and book instruction. He was a teacher.
Now, in his spare time, he was a mentor to two boys who had lost their father. He gave them their own tool boxes, fishing gear. He took them fishing, taught them how to use a saw, electric tools, everything from hammering to measuring.