There were two version of SMB 2 released. The original version, released in 1986 was a slightly graphically enhanced and even more difficult version of the first game. It was only released in Japan because it was considered too difficult for many non-Japanese gamers.
The other version is remade version of a Japan-only game; Doki Doki Panic with Mario characters.
The original SMB 2 was released in America, but it was titled "Super Mario Bros. The Lost Levels" which was one of the games included in Super Mario All-Stars.
The North American Super Mario Bros. 2 was fun but it was no where near as challenging as the Japanese version. I played the Japanese version on Super Mario All-Stars, but even with that, it's still not as challenging, since in the All-Star version, the game saves as you progress and if you got a game over, you could restart on the exacty you are on. In the real Japanese version, you probably had to go back to the very beginning if you got a game over.