Brotherhood is actually more faithful to the Manga. The first 'version' follows the Manga to some extent (about half the show) although it adds some details that do not follow the Manga.
Most of these different aspects I really don't understand.
As for the number of episodes: the 'reboot' actually catches up to the first version rather fast (in about the first 4th) and then has 'new material' (comparing to the first one, but actually it is just going according to the manga) until the end of it. Brotherhood also has more episodes (64, compared to 51) .
Also, the animation in itself is rather different in Brotherhood (much better IMO).
Basically, the first series is, in my opinion, rather poor when compared to Brotherhood.
As you've said, two animes were done:
While they both start the same way, the first one starts to develop following a whole different path than the Manga. This is because when it was being aired, the Manga was not done yet, so the plot as well as the end of the anime are invented.
The second was done after the Manga ended, so it respects the original Manga much more. I watched both because I didn't know of this distinction. But I can tell you quite objectively, that even if the first one is not so faithful, it was of very good quality, speaking of plot twists, unexpected changes in history and other similar devices.
The new Fullmetal Alchemist series is completely worth it since it adapts the original manga from beginning to end. The previous series, while being really good (in my opinion), diverges from the manga in the whole second half and ends in a way that was not supposed to be. The new series follows everything, retells in just a few chapters what we previously watched and then tells a completely new story with a completely new end.
There are a large number of differences between the two Fullmetal Alchemist anime, with far too many to list; therefore, I'll only cover the major ones.
To improve the flow of this answer, the following abbreviations will be used:
FMAM = Fullmetal Alchemist (Manga)
FMA03 = Fullmetal Alchemist 2003 (Anime)
FMAB = Fullmetal Alchemist Brotherhood (Anime)
The reason for the difference is because FMA03 was created when FMAM was still in the early stages of its development. When an anime series is based on a manga that is still in development, the anime is eventually going to reach a point where it outpaces the manga, because anime episodes tend to be developed quicker than manga volumes. When this happens, either filler for the anime is created or changes to the anime's storyline and characters are made so that the anime can continue to be developed. The latter is what happened to FMA03. FMA03's storyline started to diverge from FMAM's storyline after around 10 episodes because, at that point, the anime was starting to outpace the manga.
On the other hand, FMAB was created when FMAM was nearing the end of its development. This allowed FMAB's storyline to be more faithful to FMAM's storyline because FMAB had a mostly-complete storyline to draw from FMAM.
The major differences between FMA03 and FMAB & FMAM are:
There are quite a few differences, actually. The "original", or the "first one" is an entirely different storyline from the manga. The manga was the original thing, so most people hate the first FMA series.
A few differences between the two anime series are:
The idea behind the Homunculi
The Homunculi's appearance
Lust has a black dress in the first series, but in Brotherhood, she has a reddish-brown dress.
Gluttony, Envy, and King Bradley look the same. Sloth looks like Ed and Al's mother, while in Brotherhood, Sloth is a big (much bigger than Armstrong xD) buff male with long black hair.
In the first series, King Bradley is Pride, instead of Wrath (in Brotherhood, he's Wrath). In FMA, Wrath is a little boy (Izumi's son) with long black hair and he has Ed's real right arm and left leg. While in Brotherhood, Wrath is King Bradley. In Brotherhood, Pride is King Bradley's son, Selim Bradley. While in the first series, Pride is King Bradley.
In the first series, Greed looks normal, but
(The blonde guy in the middle is Father)
Who the Homunculi are (... meh. It's pretty much explained in the aforementioned point, so I don't think I have to re-explain).
In the first series, Hohenheim has a more round, smooth face. His glasses are bigger and more round. His hair and beard is a dirty-ish blonde color, and his ponytail is lower. In Brotherhood, Hohenheim has a rectangle-shaped head. His face has more ...umm... say "chiseled" features. His glasses are smaller and not as round. His hair and beard are light blonde, and his ponytail is higher)
In the first series, Rose's skin is brown. Her hair is dark brown with pink bangs. In Brotherhood, Rose's skin is very white. She has black hair with maroon-ish colored bangs
Rose later on
There are a few new characters in Brotherhood
I had more in mind, but in the midst of typing, I forgot what the rest was. I apologize.
Even with the absence of the other differences I had planned to provide, I hope this helped.