Basically, 教師 teach in the school or uni about basic education. In contrast, 先生 teach about something for example, how to play the guitar, how to cook, how to do something...etc. Besides politician and doctor and lawyer are also called 先生 not 教師 in Japan. And, we can use following. 先生、さようなら。 But, we can't use following. 教師、さようなら。 I hope this answer will help you.