Justice League of America is the team which features the most recognizable heroes and the team you're most likely thinking of when you think of DC. It features matinée names like Superman, Batman, Wonder Woman, Green Lantern, and the Flash, along with some lesser known heroes like Black Canary, Hawkgirl, and Red Tornado. Generally-accepted as the most powerful DC team, they are usually the ones that deal with the most dangerous, world-ending threats, though not always.
The Justice Society, in a nutshell, was the precursor to the Justice League back in the days of World War II, aka the Golden Age of DC comics. Because this is comics, a lot of the founding members are still alive and kicking, and are still running the team. It features members like the original Green Lantern, the original Flash, Hawkman, and much lesser-knowns like Power Girl and Wildcat.
It used to be that the team was just sort of a corner for the old-timers and the lesser-known Golden Age standbys to hang out in, but lately the core mission of the JSA has been to nurture and train younger heroes and provide a community -- a society, basically -- for the superhuman crowd. That's why more than half of the current members are comprised of newer heroes, legacy characters, and general up-and-comers. It's unlikely that readers new to DC would recognize most of the team. They still handle missions and stop world-ending threats and the such, but to a much lesser extent than the JLA.
There's no enmity between the JLA and the JSA; just the opposite is true. They're not one team because they're simply different teams, like the Avengers aren't the Defenders aren't the X-Men. They have different overall purposes, but often combine forces when met with threats that require both their resources.
