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I like FFA the best. I have played through Sword of Mana, but in my opinion, the original story of the old game is a little bit more..."epic". Also I like the ending of FFA better, it is "good" and "tragic" at the same time and in Sword of Mana the same scene looks rather poor.
I like the music and the graphics of the remake but the story in FFA leaves a lot more things to the players imagination when it comes to the story
But that opinion can come from my nostalgic feelings for FFA. It was the first Adventure I've ever played and the first game for my Game Boy;)
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I've only played Sword of Mana, but I chose Final Fantasy Adventure anyway because I probably would have liked it better.
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Sword of Mana is the better game. It has a more engrossing plot, the characters are fleshed out a bit better, the cutscenes are more interesting, it has a much more "Mana" feel (Elemental Spirits, Lil Cactus, Round Drops, etc). The graphics are pretty.
However, I think I might have enjoyed FFA a bit more because I'm a retro-ist at heart, and enjoyed the notion that THIS was where it all began. And it's also fun to how blatantly Square was trying to rip off the original '87 Zelda game and their own Final Fantasy franchise at the same time by combining the two into something new.