The Dragonborn is a mortal with the soul of a dragon. Dragonborn obviously have some sort of physical differences to them, as the contraption in the Akaviri temple responds specifically to your actual blood. I would say it has to do with the way the soul interacts with the body, perhaps giving it a few differing magical properties from normal. If the blood of mortals has any properties that can be used in the creation of a potion, the blood of a dragonborn is probably quite different. An interesting concept is how a theoretical blood transfusion would work, what the blood type of a dragonborn might be compared to other mortals.
In having the soul of a dragon, dragonborn typically have the disposition of a dragon, such as the indomitable want to dominate and conquer, which is why the Dragonborn were the perfect candidates to rule the empire. At least one of the people (maybe more) behind Tiber Septim was dragonborn, and conquered the entire continent using Numidium, uniting it into the empire.
Having a dragon soul also allows the dragonborn to absorb and permanently kill any other dragons, which is why they are so feared by dragons. When killed by conventional means, a dragon's soul lies dormant within its body, which is why Alduin was able to revive those killed in the Dragon War. Once their soul is devoured, they are fully gone and dead. Strangely, fellow dragonborn can be met in Sovngarde, implying the soul of a dragonborn does not stay within the body after death, perhaps a side-effect from having the body of a mortal rather than a dragon.
Dragons, and by extension dragonborn, also have an inherent ability to shout and to absorb such knowledge. A normal mortal might take a decade to learn all three words of one shout, while a dragonborn can learn it in an instant, absorbing the understanding after consuming the souls of their brethren. It's not enough to just simply know the words of power; to learn a shout is to take the understanding of the words into oneself. This is why Arngeir was wary of the player learning Dragonrend, as it was created from the old Tongues' immense hatred of the dragons.
Shouting, or the Thu'um, is a form of tonal architecture, a method of bending reality by manipulating the Earth Bones. The Earth Bones are the remains of countless Aedra that died to make Mundus, the mortal realm, sort of acting as the laws of physics for that realm. So when a dragon says Yol!, or Fire!, it is literally commanding that the universe bend to its will and create fire. The quantum mechanics of Mundus is kind of a song, played on the Doom Drum, and tonal architecture taps into this and changes the beat.
Martin Septim, from Oblivion, was also a dragonborn, and the Amulet of Kings is a massive soulgem containing the dragon souls of all the past dragonborn emperors. When Martin smashed the amulet to battle Mehrunes Dagon, he absorbed dozens of dragon souls at once and transformed into the Avatar of Akatosh.