1) 선생님 밖에 ; here 밖에 means ONLY , so only that teacher can to this task.
2) 선생님 외에 ; here 외에 means EXCEPT ; There is no one that can do this task except that teach.
So, actually two sentences have same meaning.
But, first sentence have more powerful impression that boast that teacher.
And, second sentence emphasizes that the other teachers can not to this task.
I hope you understand even though my English is very short.
In conclusion, that is the difference of stress in same situation.