Aside from the name and setting, the two game series' have little to nothing in common. Dark Alliance is a set of action RPG/ hack and slash console games with 3D graphics with completely different gameplay tone from the original. Because of that the two franchises share only a small common following and Dark Alliance is outright disliked by a large number of fans of the original.
The first game in the series, Baldur's Gate: Dark Alliance was developed by Snowblind Studios and published by Black Isle Studios. It was released on December 4, 2001 for the Playstation 2 and Xbox which got later ported to the GameCube and the Game Boy Advance. Originally there also was a PC port in production, but it was canceled before release.
The second game, Baldur's Gate II: Dark Alliance, was developed by Black Ilse Studios and published by it's chief company Interplay Entertainment. It was released on January 20 2004 for Playstation 2 and Xbox.