You are looking for accurate sound reproduction, not just acoustic range.
Sure, a cheap set of headphones may be able to do each not in the acoustic range, but what happens when it has to change sounds quickly, and constantly ? Distortion.
What you are looking for is the speaker's Sensitivity (measured in dB)... this is basically a measure of how well your speaker uses the power that is provided for it. The higher the dB rating, the better. A good rating is around 85-90dB... a Very good rating is closer to 100dB, while the best headphones on the market, may barely reach 120dB.
Beyond that, you are just looking for something that transfers the sound from the driver (speaker) to your ear with the least outside interference or loss as possible. Earbuds are best at this because they can actually pressurize your ear with bass thumps, so the bass is strong, and the sound is as accurate as possible due to lack of outside noise, only problem is, if you amplify them too much and turn up the volume, they WILL cause permanent hearing damage. Also, many earbuds simply have a large driver way out near the ear canal opening, which causes the audio they create to have to bounce around a lot in your soft ear canal before reaching your eardrum, causing a loss of high-frequency fidelity. Most of the high-quality (hence, high-priced) earbuds will actually fit inside the ear canal to aim the driver directly at your eardrum. Next best are the huge headphones that cover your entire ear. Again, they allow the driver to create pressure inside the earphone, which allows for more accurate bass, and less outside noise. Disadvantage is they are big and clunky, hence, not very portable. Then when you get to the type of headphones that just sit on your ears, you introduce a lot of outside noise, and a lack of pressure, so your audio quality will suffer.