This gets comfusing because both types of lamps have xenon gas and use an arc, but I believe that the primary difference between an "arc" lamp and an "HID" lamp is that the arc in the HID lamp is used to "excite" the gas in the bulb - as the electrons in the gas atoms fall back to their normal state they give up energy as photons of a specific wavelength. Different gas mixtures=different mixures of wavelengths, but the spectrum is still "chunky" due to the mechanism used. A simple "arc" lamp, on the other hand, makes light the old-fashioned way, like the sun - it's just really, really hot. So you get a full continuous spectrum. The Maxa beam is a true arc lamp, the Surefire beast is an HID. Because the arc lamp works by getting hot, a lot of the energy is radiated in the infrared where it doesn't do any good (unless you have NVGs) - this is why the arc lamps have poor efficencies compared to HID. On the plus side, color rendition is true to sunlight, and the arc is more of a point source than a glowing gas tube, so it can be focused tighter. This is why the main uses for the arc lamps are theatrical lighting, microscope illuminators, and searchlights. If you can get away with a little less perfect color or a slightly larger source, the HID is more efficient.
-BCK