Necessarily, not much.
For many years, I used the same shells for turkeys that I used for squirrels, in both cases, just calling them #6's or #5's.
But some "turkey" loads are designed to hold tighter patterns. And the heavier-than-lead (HTL) turkey loads will extend shotgun range a few yards over lead loads.
Regardless of what shells you're shooting in what gun or what choke, all you need is adequate pattern density for the range, and adequate energy per pellet to penetrate the target.
So long as you're shooting at a range where several pellets would hit a turkey's head/neck, your concern is adequate pellet energy. Smaller shot (such as #7 1/2 lead) generally don't have enough energy beyond about 35 yards, while #6's may be good a little past 40, and #5's past 45 (generally speaking). Going to HTL can provide a few more yards.
Generally speaking, specialized turkey loads should be designed to provide denser and more uniform patterns. By contrast, "pheasant" loads may put a premium on high velocity (which usually causes less uniform patterns). "Dove" loads may be putting the premium on "cheap", maybe using less costly components and less quality control.
Some "turkey" shells, such as the Federal Flight Control, have a special wad that tends to hold the shot together better, especially when used in factory full or more open chokes.