I'm no expert, but I'll give your question a stab.
Hevi-Shot was developed by Environmetals and the name "Hevi-Shot" is tradmarked by Environ. Another trademarkedname for "heaver than lead" shot made by Environ is Hevi-13. Initially Environ didn't actually load and sell the ammo, but they entered an agreement with Remington to load the shells. They were sold by remington as "Hevi-Shot Magnum" turkey loads and the box carried the Hevi-Shot logo and trademark. Sometime in 2006 Environmetals suspended the agreement with Remington and started loading and selling their own brand of ammo trademarked as "Hevi-Shot" loads.They sell loads for not only turkeys, but waterfowl, coyote, pheasant, etc.
http://www.hevishot.com/products.html
Environ also sells Hevi-Shot to reloaders. The Nitro Companyn Mountain Grove, Missouri purchases Hevi-Shot from Environ and loads it in their own propriatary line of shells which we commonly call "Nitros". These are extreme, max loads and cost around $5.00 per shell. These are the shells I have previously spoken about in this and other threads. They also load a "econonmy" line of turkey ammo containing a triplex combination of copper coated lead 4s and 5s with Hevi-Shot 7s.
http://www.nitrocompany.com/
Confusion exists concerning the Remington line of "Nitro" ammunition and the Nitro Company's line of ammo we commonly call "Nitros". Remington has been loading various lead shells in their "Nitro Premier" line for years and owns a trademark on the "Nitro" name as it applies to shotshell ammunition. These are not the "Nitro" loads which turkey hunters are referring to. The Nitro Company doesn't refer to their own ammo as "Nitros" because it would be a trademark infringement. Turkey hunters simply refer to the Nitro Company's turkey loads as Nitros because it is easier to say than "the Hevi-Shot turkey shells loaded by the Nitro Company"
Other manufacturers,specifically Winchester and Federal, load their own version of "heavier than lead" shot. Winchesters "Supreme Elite" line has the "Extended Range HD" turkey loads containing Winchester's brand of "heavier than lead" shot. They claim that theirs is better because it is rounder than Hevi-Shot (Hevi-13).
http://www.winchester.com/products/catalog/shotlist.aspx?bn=16&use=16
Winchester HDis rounder but I don't know if it makes any difference. Hevi-13 actually isn't perfectly round - the pellets have a nipple sticking out but it doesn't seem to affect performance.
Federal loads their version of "heavier than lead" shot they call "Heavyweight"in the Premier Ultrashock Heviweight line. I don't have any personal experience with it but the 3.5" 12 gauge and 10 gauge loads of straight 7s are popular with many hunters.
http://www.federalammo.com/ballistics/Ammo_Search.aspx?act=choose&firearm=5&s1=1
There you go - that's about all I know.