I think a usable, if not universally true,
tl;dr: Low impedance (less than ~100Ohm): mobile use. 250Ohm and up: home / studio use.
According to Beyerdynamic the main reason for producing high-impedance headphones is that the wire used to create the voice coil: thinner wire means less weight (which is good as less moved mass is easier to accelerate in the magnetic field) but higher resistance, thicker wire (larger diameter) means more weight but lower resistance, and, somewhat related, impedance (the nominal impedance is normally measured as 1kHz). The details of that relationship are a bit complicated to say the least
Beyerdynamic states that their higher impedance versions (they produce several models in variants with different nominal impedance) generally sound 'more natural' for what that may be. I'd say they have a bit smoother treble.
Beyerdynamic FAQ
Higher impedance means the amp must have more voltage swing to drive the headphone, something that is hard to achieve on a portable amp, fed by a 5V battery or something like that. There are exceptions of course, e.g. the Meier Audio QuickStep uses a 9V battery and a symmetric power supply derived from that as +/- 9V, the CEntrance Hifi-M8 uses +/- 10V. Both should be able to drive high impedance headphones.
High impedance headphones also have the benefit of pairing well with OTL tube amps as those have a rather high output impedance that might affect the sound of low impedance headphones negatively.