They're both "baby" studs/spuds/pins... up to a point. Notice that the one with the 1/4-20 threaded top section is about the same overall length, but that the 1/4-20 threaded section is just a touch skinnier than the 5/8" body below it.

A normal baby socket is at least 1 3/16" (30mm) deep. The socket on your flash bracket is that deep - your 1/4"-20-topped stud wouldn't fit all the way if it weren't - but it's not 5/8" in diameter for all of that depth. The 1/4"-20 bit at the top can go places where the flap-topped pin on the C-stand can't go. (You would probably find the same bad fit if you put the standard plastic thread protector over the 1/4"-20 thread on that stud.)

So the problem is the flash bracket. Don't get too upset about that - it's both good enough for "normal" use (most light stands in use aren't "proper" grip equipment; they're lightweight "location" stands with the sort of studs your smaller stand has, including a 1/4"-20 threaded section or reversible 1/4"-20 and 3/8"-16 sections) and cheaper to manufacture and to purchase with the socket it has. The socket can be fixed in a couple of seconds with a drill bit that can flatten the bottom of the socket - a brad point drill should do the trick. And any "pro" lighting equipment will already have a socket that's shaped properly for a standard baby pin, since they're more likely to be regularly used on C-stands, plates, grip arms, and so forth.

Oh, and the extra length below the reduced area? That's to allow you to attach grip heads and so on - they need a much longer 5/8" pin than a light's socket does.