Complete, faithful adaptation of the finished manga — the version most commonly recommended and referenced as a genre landmark.
Follows the manga for roughly the first third, then diverges into its own original, darker ending. A distinct story, not an inferior copy.
The core alchemy rule ("to obtain, something of equal value must be lost") drives both versions, with Brotherhood building its full arc around testing it.
Most viewers are advised to start with Brotherhood since it's the complete, faithful story — the 2003 version is better appreciated afterward as a distinct alternate take, not a prerequisite.
The 2003 anime was produced while the manga was still being written and had to create its own ending; Brotherhood was made after the manga concluded and adapts it faithfully in full.