As with electrostatic speakers, the main difference is fidelity. To be clear, by fidelity, I mean less change to the recorded sound.
TLDR: It’s impossible for any other means of making sound to alter the sound less than electrostatics do. The result is the most convincing, detailed, natural, clear music possible. The knock-on result of that is you'll be drawn into the music more so than with any other type of headphone. In short, electrostatics have the greatest potential for making you feel directly connected to the musicians and singers, like being a witness to each actual performances.
Now for a technical explanation of the differences and the effects they have.
Dynamic drivers are asked to do the impossible, namely act as pistons, when they're really a diaphragm being pushed from the middle by a much smaller, non-linear motor. The diaphragm thus moves in many ways, but not quite like a piston.
For one thing, it develops multiple resonances, and different areas move at different speeds and in different directions. This is true for the surround, the part that suspends the edges of the cone, moving quite differently than the diaphragm, yet far enough to make sound, and the diaphragm breaking up at certain frequencies into separately resonating areas. This causes unnatural ripples in the frequency response, meaning the sound levels of some tones are not the same as when they were recorded.
The combination of motor and diaphragm also has substantial mass that has to be accelerated. That causes the sound to be emitted with a delay relative to the signal, and this delay is proportional to the frequency. This misaligns the fundamental and harmonic content of the signal to various extents.
Also, most of the energy goes into accelerating the mass and only a little into making sound. Not that this is very important under most conditions, but when a dynamic driver is playing loudly, its coil heats up and thus develops additional electrical resistance, and this reduces the sound level out of proportion to the signal level, a form of distortion called power compression that reduces the naturalness of the musical dynamics.
Lastly, the non-linearity of the system causes harmonic and intermodulation distortion. There are two sources of non-linearity. One is the variation in diaphragm force vs. position. This causes both kinds of distortion, but mostly harmonic distortion, namely adding harmonics that are not present in the signal, but are at least harmonically related to the signal. The other is the variation in inductance with diaphragm position, which causes the phase of the sound to vary with movement of the diaphragm. This causes intermodulation distortion, that is, the development of sounds that are not harmonically related to the signal, and these are particularly harmful to the naturalness of the sound.
In a high quality, over-ear, dynamic headphone, the driver is usually 40 mm in diameter, sometimes 50 mm, and the opening of the headphone can is usually in the range of 600 x 800 mm. This means the sound can vary substantially depending on how the headphones are placed on the head, because the alignment of the driver with the ear canals and pinnae can vary. There are other influences on the sound with position, but this is the big one.
Before we get to electrostatics, I should mention another type of driver, the large area, planar magnetic. This is a special type of dynamic that solves many of the problems. It has conductors bonded to a thin diaphragm, and current through those conductors causes the diaphragm to move in a magnetic field.
The large area makes the sound more independent of position on the head, and the membrane gets more or less even force over its area, so it moves more as one, and it’s far less prone to breakup. Their inductance is nearly constant, so intermodulation distortion is minimal. These are the technical reasons why good ones sound better than standard dynamics, at least to people who recognize and appreciate high fidelity.
There is still significant mass on the diaphragm, however, because of the metallic electrical conductors on the diaphragm. The diaphragm itself must be thick enough to support those conductors and withstand the heating that occurs as current flows through those conductors. The heating varies the tension of the diaphragm, because the plastic expands when heated, and this varies its sound, so in addition to power compression, the frequency response is somewhat loudness dependent.
Finally, we get to electrostatics. There are two kinds: those with the diaphragm charge embedded into the diaphragm, called electrets, and those that have an external polarizing supply, called true electrostatics.
Electrets are impractical, mainly because they are inefficient. They are inefficient for two reasons. 1) The highest voltage charge that can be embedded is about 600VDC, which is quite low, and efficiency is proportional to voltage. 2) The membrane materials that can hold embedded charge are thick, and this causes the sound to roll off within the audio band.
Also, it’s impossible for the materials that can hold embedded charge to maintain the tension needed for reproducing a wide range of frequencies. They consequently don’t start adding sound until above about 8000 Hz.
As a result, makers claiming that their electret headphones have “electrostatic technology” in them are technically correct, but at best, they are selling dynamic headphones with the very top octave or two supplemented by the electret. An increase in the fidelity of the upper harmonics makes very little difference in the sound we hear, so even if an electret were loud enough to keep up with a dynamic driver, it still wouldn’t matter.
True electrostatics offer many advantages over dynamics.
There are two most important aspects. One is that the diaphragm material is so thin and light that it presents no mass loading within the audio band. All energy applied goes into making sound. The diaphragm acts as a virtual air barrier and vibrates to make sound without adding any voice of its own. The other aspect is that the force is spread evenly across its surface. Put together, these aspects mean a true electrostatic has no choice but reproduce the signal with the highest fidelity possible -- no breakup, no resonances, no frequency-dependent inductance.
A true electrostatic is also quite efficient. The diaphragm in a true electrostatic is polarized to the limit of what the air can support. This makes them at least 9 dB more sensitive/efficient than an electret, all else being equal, and they can keep up with a dynamic driver.
The diaphragm material can maintain tension indefinitely, and can thus be arranged to reproduce many octaves, even from a headphone sized transducer.
Since ours are or will be the only true electrostatic hybrid, I hope no one will mind that I phrase the rest of this with our headphones in mind.
Ours cross over to the woofer at about 800 Hz, meaning the range of sound that requires the highest fidelity is produced by the highest fidelity transducer possible. Running the woofer only below the frequencies where its diaphragm would break up means its fidelity is also quite high within the range it is being asked to run.
The area of our electrostatic transducers is nearly the entire aperture of the headphone, so they are relatively immune to variation in sound from differences in how they fit to the head.
The crossover from electrostatic to dynamic must not create an evident disjunction in sound, and we have accomplished a seamless transition. In fact, compared to full range electrostatic headphones, ours go deeper and have a richer sound in the bass, thanks to the use of a dynamic woofer.
One main reason we use a dynamic woofer is to be able to build everything into the headphones, namely the polarization supply and step-up transformers. Because full range electrostatics have large step-up transformers to cover the bass, these items must reside in an external box that gets plugged into wall power. This makes ours the only true electrostatic that's portable. The other main reason is to make them sensitive enough to drive from a phone or portable player at levels loud enough to use outdoors, so unlike full range electrostatics, they don't need an amplifier.
I hope this was helpful and not overly self-promotional.