Basically FMAB is a full adaption of the manga while the first anime was only a partial adaption.
The first anime was made when the manga was only maybe 25% complete, so they adapted the first bit of the manga into an anime and then once they ran out of source material, they made up the rest of the story as they saw fit.
Brotherhood was made when the manga was like 99% complete and thus follows the manga infinitely more closely.
Brotherhood is much better once you give it a chance, all too many people don't watch it just because the first bit is similar to the other anime. Around where Greed isintroduced is where the first anime split off from the manga, so it will be there that you get purely new material. I'd still recommend watching all of it though, Brotherhood is just plain better, anybody who argues against that clearly has no idea what they're even talking about.
Most people will say they enjoyed the first anime more because it had a much darker theme, which is what makes me think they're morons. Brotherhood has a good bit of comedy mixed in, but it's much darker in nature than the first anime. Also the villains are actually relevant, in the first anime the Homunculi had virtually no essence, and Dante was a worthless final villain.
At any rate, watch all of Brotherhood, even if you know the plot of the first 10 or so episodes, you can still easily enjoy it since it's new animation, new dialogue and voice acting, new music, and a fresh atmosphere.
Also has kickass opening and ending credits.
EDIT:
Oh yeah and Brotherhood has Greelin, really awesome character.