benutzen (more commonly: verwenden) is used for objects you can touch, while nutzen rather applies for ideal, abstract or non-physical things. You use duolingo to learn German – Du nutzt duolingo, um Deutsch zu lernen. To use an unique opportunity – Eine einmalige Chance nutzen. Wir nutzen den Lehrer seems incomplete, since it doesn't obey this rule and how would you use a person? Person A taking advantage of/exploiting person B (with a negative connotation) would be benutzen or ausnutzen. Wir nutzen das Wissen des Lehrers, um unsere eigenen Kenntnisse zu erweitern – We use the teacher's knowledge to extend our own proficiencies, this would be legit.
Thank you! I've been trying to figure out the distinction for awhile now--this is a great distinction. I was also wondering whether "nutzen den Lehrer" should translate a little less uncomfortably, but wasn't quite sure how to ask. Thanks for that explanation, too.
Thank you so much for the explanations! Nevertheless, given the fact that my level is still basic, I am not quite sure of what the "um" stands for in your cases...
http://de.pons.eu/dict/search/results/?q=um&l=deen&in=&lf=de Scroll down to III and "um etw. zu tun."