Twist Bits
Twist bits get their name because they look like they've been twisted to get the flutes or concave shafts that wind along the bit. The flutes are fairly deep on twist bits. They are designed so the cutting edge cuts into the material being drilled and the shaving is "cleared" by passing up through the flute were it can fall away from the work. Without the flutes, the drill wouldn't clear. It would jamb and overheat.
Other Bits
Drill bits come in a broad variety of designs, whereas the designs of burs are similar; it's their shape and size that varies. There are auger bits and paddle bits and all different lengths and diameters. Not all bit designs are similar to burs, but they all function similarly to twist bits: cutting with the tip, then clearing with the flute or some other kind of channel.
Industrial Diamond
There are many popular grades of hardened steel alloy used for machine tools such as bits and burs. There are also specialty alloys with high concentrations of ultra-hard metallic elements including tungsten and cobalt. Another way to enhance a cutting edge is the addition of industrial diamond. Powdered industrial diamond is coalesced with molten metal or with a coating material. The diamond-impregnated metal or coating stands up much better to frictional wear.
Burs
Both burs and twist bits are sometimes diamond-coated, though their cutting edges are different. While twist bits shave metal from the tip of the bit in a deeper scooping action, burs shave away small metal filings from the side of the bur. Most bur tips are not a significant functional element as they are in a drill bit. Bur shafts are the primary functional element.