Just like $2\times3=3\times2$, There is no difference between newton-meters and meter-newtons. They're two different ways of saying the same thing.
Probably your book is trying to avoid confusion when you learn about energy, which is also measured in newton-meters, although we normally rename the unit, when referring to energy, as joules.
You should go ahead and call the unit of torque "meter newtons" in your class, because that's what your instructor expects, but. But be prepared to see other people call it "newton meters". I'd even strongly recommend using newton-meters anywhere except in your class, since the unit ${\rm m\cdot N}$ (meter-newtons) is much too easily confused with $\rm mN$ (millinewtons).