"The teacher can speak many languages."
I don't think this Chinese sentence is correct. 能 is about whether circumstances allow you to do something, regardless of whether you have the skills to do so. 会 is about whether you're able to do something, regardless of whether it's possible in this circumstance. 会 should be used in this sentence.
Both 能 and 会 can be used in this sentence and have basically no difference.
There are some differences between 能 and 会, but not quite as what you said.
There are slight differences between 能 and 会. 能 indicates possibility (可能) whereas 会 indicates ability (可以).
It should be 会 here as it is an acquired skill, buuut since we have the verb 说, 能 is not completely inadequate. Honestly, I'd say "老师会很多种语言。” but I'm lazy so...