Which one you should watch?
Both Full Metal Alchemist and Full Metal Alchemist Brotherhood are really good anime series adapted from the manga of the same name. For us we watched Full Metal Alchemist (2003-2004) initially, and were quite pleased with the story line, art and vivid background and effects. Our curiosity, wasn't answered with what was presented in Full Metal Alchemist.
To overcome this, we read the manga by Hiromu Arakawa which was by far a masterpiece, the emotions, relationships, unanswered questions that were now answered and even the introduction of new characters that were never present in Full Metal Alchemist. Thus, we had reached euphoria with it.
Then in 2009, Full Metal Alchemist Brotherhood was released. It did not fully capture the essence of the manga but it was closer to the manga than Full Metal Alchemist (2003-2004) series. The Brotherhood features considerably advanced and better animation than the original. Brotherhood is more alive with details in everything and backgrounds are more dynamic and elaborate. The main antagonist for Brotherhood was the same as in the manga, in comparison to FMA 2003-2004 version, it was a woman by the name of Dante. Hohenheim, played a bigger role in the Brotherhood series, and a backstory of how the antagonist in Brotherhood came to be. The ending for both series were vastly different, but Brotherhood followed through with the manga where Edward sacrificing his ability to use alchemy in exchange for the body of Alphonse.
While both the anime series are a great watch, considering they are derived from one of the greatest manga anime series of all time, they have their fair share of differences. This is our personal opinion but we always recommend that new anime lovers or manga lovers watch/read Full Metal Alchemist Brotherhood as it follows the original events of the manga so its story line coincides with the manga’s story line.