Your voice does in fact sound different to other people than the way you hear it when you speak -- it's a real thing, not just some matter of subjective perception like being too used to your own voice that it becomes distorted when you hear it played back from a recording device, for instance. Why you can never hear your own, real voice without assistance (recording yourself) has to do with how sound reaches your inner ear. Basically, your inner ear picks up acoustic vibrations like the chirping of birds, rattle of the city or people's voices and translates these vibrations into electrical signals that the brain can process as "sound". The inner ear, however, also picks up vibrations conducted by the bones in your neck and head. This combination of internal and external vibrations produces an uniquely characteristic voice which you won't ever be able to hear elsewhere!
Image: FACTS WT
A voice literally inside your head. More than one calls for a psychiatrist's appointment.
A person's voice is basically sound, which in turn is vibration that propagates through air. Humans are able to speak thanks to a nifty biological gadget called the "voice box": cartilage casing in the throat that is often referred to as the Adam’s Apple” (for men), the “Eve’s Apple” (for women). It's here that the twin infoldings of tissue (roughly) the size of our eyelids called the vocal folds reside.
Diaphragm action pushes air from the lungs through the vocal folds, producing a periodic train of air pulses. This pulse train is shaped by the resonances of the vocal tract, and since each “voice box” is shaped uniquely so will the voice. Since men have an Adam’s Apple, their voice is lower since the vocal cords are longer. The basic resonances, called vocal formants, can be changed by the action of the articulators to produce distinguishable voice sounds, like the vowel sounds.
This sound energy is spread throughout the air, and if a human is nearby with a pair of healthy ears, this energy will hit the cochlea (the inner ear) through the external ear. Fluid movement (air pressure waves) through the inner ear causes changes in tiny structures called hair cells. As these hair cells move, electrical signals from the cochlea are sent up the auditory nerve to the brain, which is then converted into information we can commonly refer to as sound. Hair cell loss comes from noise exposure, aging, toxins, infections, and certain antibiotics and anti-cancer drug.
That's how humans are able to speak (have a voice) and hear (listen to the voice).
At the same time though, those same vibrations that resonate inside the voice box get conducted by bones inside your body and reach the cochlea directly through the tissues of the head. If you cover your ears, you'll still be able to hear your voice, simply because your head is ringing. The tissue is better at transmitting low frequencies than high ones, which makes you think your voice sounds lower than it does to other people.
Ultimately, the voice you hear when you speak is the combination of sound carried along both paths. Some people are more sensitive to vibrations conducted through the bones. In extreme cases, these people might hear the sound of their own breathing and even eyeballs moving in their sockets. Some people have abnormalities of the inner ear that enhance their sensitivity to this component so much that the sound of their own breathing becomes overwhelming, and they may even hear their eyeballs moving in their sockets.