To add to this, Akira Toriyama is not only the lead designer of DQ and creator of Dragon Ball, he also is the lead designer of games like Blue Dragon, Tobal Nº 1 and Nº2, and the highly praised Chrono Trigger, Square had a dream team in that game:

Hironobu Sakaguchi

Yoshinori Kitase

Yasunori Mitsuda (lead composer of this game, Uematsu was his mentor, even recommending him at Square and helping him with this game)

Nobuo Uematsu

Takashi Tokita

Akira Toriyama

Yuji Horii

Masato Kato (Known before this for his work on the Ninja Gaiden NES games)

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Now, back to Dragon Quest, here are the original boxarts, they're another reason why DQ games are so popular in Japan

DQ FAMICOM boxarts down

As opposed to the US boxarts that were really lame, at least the Megaman boxart artists made us laugh...

It depends on your tastes.  The way I like to answer that question is by describing each series strengths.  Final Fantasy focuses on Story and going for the 'cenematic feel' (aka trying to be like a movie) as described by series creator Sakaguchi.  Dragon Quest on the other hand focus on the roots of the RPG series, focusing on exploration, NPC interaction, item collection and gameplay, as described by series creator Yuji Horii.

If you like games that are more open-ended, with more exploration, a higher challenge, emphasis on keeping your characters well equipped and a deeper combat engine, you'll like Dragon Quest.

If you like games that focus more on story and long dialogue sequences, let you know when major bosses are coming, lay out the path you need to take and have a more 'active' or 'free' battle engine, or frankly if you're just looking for a game more like Final Fantasy, then this isn't a game for you.

Pokemon says hello...