One model, with three faces the Beyerdynamic DT770 Pro Headphones has been crafted in various iterations, each with a specific purpose in mind. At 80 Ohm this version sits right in the middle, between the 32 and 250 Ohm models, and is perfect for recording, mixing, and general applications. The headphones are especially useful for front of house and drum monitoring, as their lower impedance produces a louder sound. Furthermore they have been designed for extended periods of use, meaning they will be comfortable for the entirety of your set or studio session.
With its thicker wire providing a lower impedance, more signal is let through, though the extra weight of this wire does affect the diaphragm, which cannot move with as much accuracy. On the flipside, this lower impedance means the 80 Ohms Beyerdynamic DT770 Pro Headphones are louder and require less power to provide more volume than its high impedance sibling. With this capability these headphones are perfect for battery-powered mobile applications, or where more volume is needed and some loss of quality won't be noticeable, like front of house performances.
The difference between open and close back headphones can be confusing. One is not simply better than other, it's all about application. For recording, you will typically want a pair of closed back headphones, as they retain all the audio that is being captured, with no spillage that could be picked up by your microphone and affect your recording. Using these for mixing is less advisable, as, due to their construction, they tend to lose bass response.
For mixing you would generally be best off using a pair of open back headphones, as they have a clear bass response, with a natural sound that can pass more freely through them, and they are generally more comfortable. These would not suit recording however, as they spill audio easier and this can be captured by sensitive microphones.
Rather than purchase two set of headphones, the Beyerdynamic DT770 Pro Headphones (80 Ohms) has your answer in one. Although closed back in their design, these headphones utilises Bass reflex technology, along with larger ear cup sizes and velour ear pads, to offer an improved bass response, bringing the nominal frequency response to 5Hz-35kHz, whilst retaining the qualities of being hard back.