Compared to the original version of Doom3, dhewm3 has many bugfixes, supports EAX-like sound effects on all operating systems and hardware (via OpenAL Softs EFX support), has much better support for widescreen resolutions and has 64bit support.
Note that while the Doom3 source code has been released under GPL, you still need to legally own the game and provide dhewm3 the game data to play.