Fullmetal Alchemist : Brotherhood is nor a remake or second season of the original Fullmetal Alchemist.
Reason it is not the second season is because (2003)'s Fullmetal Alchemist DID end. There was an additional movie called Conquerer of Shamballa. This movie officially concluded it.
Not a remake either because there has never been ONE anime to date that has been remade by any other company, franchise, etc. As much as some companies would love to remake some of their epic fails, they do not feel like spending the kind of time and money they could to make a new shitty anime. Quality over quantity my dear.
It's well damn obvious the original series was rushed. As Death the Kid quotes, "Everything not symetrical is ugly". My sentimentals exactly. The original Fullmetal Alchemist had a completely random number for number of episodes (51 to be exact) and is grotesque in telling the story.
Compare and Contrast of the two:
Fullmetal Alchemist : Brotherhood : It follows the manga with less action and blood. There is no gore in this series (not noticed any yet), which means piles of gushing. Less action, more story.
2003 Adaption : It was made loosely with an oddly thought conclusion, more action and lots of gore. Less story.
Fullmetal Alchemist : Brotherhood :Tells the story with no fillers in a quick, yet detailed way.
2003 Adaption : Tells the story in a completely scattered, exagerated, and boring way.
Fullmetal Alchemist : Brotherhood. : The color scheme has also been redone with a nicer look.
2003 Adaption : Lots of color and shading. Not colored as well as Brotherhood.