Pros and cons of high impedance headphones:
Headphone impedance is usually increased by thinner wire and most importantly more turns of wire in the voice coil. More turns or loops creates a larger field (area of magnetic influence). In layman's terms more magnetic force for the coil to move the diaphragm. Thinner wire usually works out to a lighter, more responsive diaphragm. Depending on the headphone design, this may lead to more accurate response.
The displacement (amount of movement) of the diaphragm (the part that vibrates to produce sound) can be better controlled via a more accurate flux (magnetic field to pull and push the diaphragm).
Difficult to drive for small headphone amps with low output voltage and low gain.
Most high impedance headphones need an amplifier with higher voltage gain and higher output voltage, e.g. the 600 ohm Beyer DT770/880/990 series.
Allow Solid State Op Amps to work more efficiently with less distortion. Have a look at Op Amp data sheets and a graph of distortion vs. output impedance for most audio Op-Amps and you'll get the idea. This is a very complex subject, but most Op Amps are designed to output (typically) up to 10 Volts into 600 ohm loads or higher.
Pros and Cons of low impedance headphones:
Headphone impedance is usually decreased by thicker wire and less turns of wire in the voice coil. The magnetic field is built up by more current.
Easier for small and/or portable headphone amps to drive. For example: an iPod or MP3 player headphone jack. Many small and/or portable headphone amplifiers are designed to output a volt or two into low impedance, high efficiency headphones, e.g. Grado headphones.
Low impedance, low efficiency headphones usually sound better when driven by a desktop amplifier, e.g. Audeze LCD-2 or AKG K70X.
Low impedance headphones usually sound better when driven by a solid state or a transformer coupled vacuum tube amplifier. Low impedance headphone do not usually work well with Output Transformerless vacuum tube amps.