IN most cases, not a lot.... and a lot of braids marked "casting" work just fine for jigging, and those marked "jigging" work just fine for casting...
Typically your jigging braids are often multicoloured, a different colour every 10 metres repeating every 5 colours... this is great for jigging in water where you are marking fish at a specific depth mid water or deep water and can then count the colours down to the specific depth so you know you are "in the zone" rather than working your jig through a whole bunch of "dead" water before getting to the fish...
Some of the higher end casting braids ( brands like Varivas etc ) are specifically designed more for casting in that they are softer/limper and do seem to cast better than the jigging versions...
That being said, another high end brand, YGK Ultra has their Jigman and Castman ranges, which are identical lines, just that the one is coloured coded, the other not...
But for the most part it seems the jigging dedicated braids are slightly less limp...
Alot of braids out there work well as multipurpose lines... brands like Jerry Brown Hollowcore for example make exceptional casting and jigging braids...
Personally I don't feel there is a hell of a lot of difference in most, but the higher end casting braids certainly do cast better.... I'm fishing Varivas GT Max SMP on one of my heavy set ups at the moment and it cats like an absolute dream...
