Dragon Quest is more popular in Japan than Final Fantasy, in the west they never had a chance...
1. The FF games came as Final Fantasy and the DQ games came as Dragon Warrior (until DQ8 on the PS2)
2. Square was more organized than Enix in the West.
3. When Square fought with Nintendo, Sony came along with a killer West plan that benefited both companies, the Playstation becoming the Nº1 brand on video games, and Square becoming the Nº1 JRPG company worldwide. Meanwhile Enix never did anything to expand Dragon Quest outside Japan, that was until the merger.
4. When Square Enix was founded, the changes became visible for the DQ franchise, they took the original name back in the west and dumped Dragon Warrior, and they shipped Dragon Quest VIII with the Final Fantasy XII demo, reaching a wider audience, and now they have an even bigger opportunity, DQ9 is on the DS and DQ10 is on the Wii, both Nintendo consoles and both widely popular, and Iwata promised to promote DQ in the West...
Of course, Pokemon beats both of them in every region... But that's another story...