The JSA, are much older than the JLA. Back in the old comics, when the 2 teams joined it was pretty rare. The JSA are from Earth 2, the JLA are from Earth 1. They're a lot like the JLA, only older and with different costumes. Like Green Lantern. His name is Alan Scott and he is the first ever GL. His costume is much different and he's old. Or the Flash(Jay Garrick). Costume is different and he's an older type guy.
The JSA dosen't have their own version of Superman, but they kinda have a Batman. Wildcat. He's a boxer and he wears black. Now the three members I've just mentioned would be considered their "Big 3". They used to fight crime on their world, until the multiverse collapsed into one single earth. Both teams have tremendous respect for each other, and like the JLA their line-up has changed frequently. Hawkman is a major member now. I think that's the main explanation. Hope that helps a little.