First, Adam's world is not the same as the world that will be when Moshiach comes.Adam had a job to do - not to eat from the Eitz Hadaas - which, had he fulfilled it, moshiach would have come. But he didnt, so now we are working on returning the world to where Adam was, so that at that point, we can take a second shot at succeeding where Adam failed. Thus, the Geulah has 2 parts: part 1 - where moshiach comes and we return to the point where Adam was, and then, when we accomplish what Adam was supposed to, and rise even higher, the Final Reward comes.
Second, Adam did not have a Yetzer Horah - that was the Nachash - and he was warned not to let it in to gan eden - לעבדה ולשנרה - to "protect it" from what? From letting the Nachash enter! Adam thought, though, that Hashem would appreciate it more if he did get the Yetzer Horah and then beat it. So he let the Nachash in.
And then it beat him.
Lesson learned: Keep the Yeter Horah outside, even if you think you'll do more for Hashem by letting it in.