RE: what is hevi shot?
Hevi-shot itself is a high-density, non-toxic pellet made from tungsten, nickle, iron and one or two other alloys. It was invented by a waterfowl hunter in the NW part of the US in the late 1980s/early 1990s and gained federal non-tox approval sometime around 1993-95. Hevi-shot itself is a proprietary product of Environmetal, the company is based in Sweet Home, Oregon.
Hevi-shot is also a generic term for "high-density" shot, which is basically any type of shot that has a density greater than lead. Density of shot is measured using grams/cubic centemeter or g/cc. Lead has a density (depending on the source) of about 10.8 g/cc. Steel is typically around 7.8 g/cc. Hevi-shot as found in the old Remington loaded shells, and now the Hevi-shot "goose" loads are 12 g/cc... hence they are "heavier/denser" than lead. Hevi-13 is also a product of Environmetal and was developed with turkey hunters in mind. It has a density of (suprise suprise) 13 g/cc. Most of the major ammunition companies now have some proprietary derivitive of hevi-shot in the form of their own High density products.... Remington now has Wingmaster, which is very very similar to Hevi-shot, only slightly softer and more dense. Winchester has Extended Range HD which is almost the same as Wingmaster (although winchester brought out Ex Range before Wingmaster hit the market), Federal has heavyweight, which is acctually to my understanding the most dense of any of the mass produced/mass loaded shot, with a density approaching 15g/cc. There is also a product that has been out for a few years now call Tungsten Super Shot, which is 18g/cc. There have been reports of duck hunters using 28 gauges with cylender chokes taking ducks cleaning out to 50 yards with #7 TSS. Its definately impressive stuff. There are lots of other variants as well... this is just some background.
The reason you should not shoot hevi-shot or other high density shot out of a lead only choke tube is because tungsten, like steel, is a very hard shot. You can scratch you bore or your choke tube if you over choke the shot. Most of high density shot charges perform much better with slightly more open chokes. Which is why you see many of them marketed as such.
As r33h wrote, the advantage is being able to shoot smaller shot, which will give you a more dense pattern, and often greater penetration. The more chances you have to put pellets in the spine/brain of a turkey the better your odds are, but you do still need to have enough energy to penetrate the bone and get to the vitals (the spine and brain).