First,教师 is more official than 老师.for example we can say, 作为一名教师，他尽了他应尽的义务。他想当老师。if you say,他想当教师，that is to strange. Second, when we say hello to teacher,we will say, 老师好，we don't say 教师好。 I hope, that is helpful for you.

老师和教师，并不是用official、formal/informal、written/spoken就可以解释清楚的。这是一个有趣的语法现象，一些语法学家也认为是一个不太好解释的现象，建议你还是跟着母语者学吧，他们怎么用你就怎么用。

教师 is written, 老师 is spoken.

Actually they are the same meaning but used in different situations. 教師 is more like a position. EX : 教師資格(qualification of teacher)、教師工會(teachers union) But in daily life speaking, we use 老師 more. EX : 老師好、老師早安、王老師(teacher Wang)

教师 tend to be more formal while 老师 is more colloquial. Chinese seldom use 教师 in spoken language (For example, when talking about some teacher, Chinese people use X老师 rather than X教师. Especially, when you talk to a teacher in person, you must call him/her 老师). However, in many situations you can substitute 老师 for 教师 in written language. And there are basically no other differences.