Unlike the Justice League members, in the comics the Teen Heroes were allowed to grow up. It's weird logic as no one else in comics age, but whatever. SO the original Titans (Robin, Wonder Girl, Aqualad, Kid Flash, Speedy and so on) actually grew up and got new alter egos. Robin became Nightwing, Wonder Girl became Troia, Aqualad become Tempist, Speedy became Red Arrow.. and so on. When Wally "graduated" be took over as Flash when Barry Allen died. He joined the Justice League in Barry's place but also was a member of the graduated Titans. Keep in mind that while these kids were aging, Justice League books were going on at the same time, and their heroes weren't aging. It's just how these things work.
In animation, the Teen Titan's show technically didn't exist in the Timmverse (even if the shows were created at the same time by the same people). The styles were different and the creators of Titans didn't want to have crossovers with the Justice League people (mostly Batman) because they felt those big personalities would trump the Titans. Speedy showed up as a cameo, yes he's the same character as he was in Teen Titans, but it wasn't the same universe. The reason Wally West was Flash in JLU instead of Barry Allen was because Wally was the current Flash in the comics at the time, plus Wally is funny while Barry is a bit of a stick in the mud. They needed a comic character so they picked Wally.
"the creators of Titans didn't want to have crossovers with the Justice League people (mostly Batman)" funny you should say that since batman was the only one to clearly reference the titans in his static shock crossover. actually, does that mean that batman: the animated series takes place years before justice league?