The video stops where it does because Tetris doesn't get any faster after a certain point. If a player is good enough, he can just keep playing at Level 20's speed indefinitely, which makes basing a world record on just score rather difficult. More modern Tetris games, like Tetris Party on Wii/DS, have a marathon mode that ends at Level 15, so high scores are based on who can get the most points in 150 lines.
Since only 3 or 4 people have been recorded as maxing out the score on Nintendo Tetris, Buco is tied with them as far as the score counts, but he's considered having broken the record because he did it with fewer lines than anyone else has, which makes him the "better" Tetris player.