My take:
Final Fantasy - Stories about darkness that emphasize the goodness (or in this case light, literally) in life. Stories come across as deep with some macabre leanings but are generally (sans X) simplistic decidedly aimed at teenagers. They're games with dark settings that have a light hearted tone. Kefka kills people mindlessly but the game writes his dialogue as a bad, cheesy joke. Cloud has to save Tifa from a life of prostitution but not before dressing up as a woman. In X, it's even literal because Yuna is setting off to die and she wants the journey to be full of "laughter". Continuously changes gameplay, but while this can be a positive, it's also Final Fantasy's biggest weakness. Meaning, you can't really
rely upon FF to be good because it's largely inconsistent. You might love the gameplay and tone of FFXII, but guess what, you're never going to get a follow up and they'll never bother to follow up on what made it popular. That kind of stuff burns, especially when you have sequels like XIII which downgrade the previous games accomplishments. FF games do not build upon each other gameplay-wise and this is why each game has its detractors and critics as well as fans and lovers.
Dragon Quest - Stories about goodness that emphasize the darkness in life. DQ is deceptively cute. The above poster clearly hasn't played too many DQ games.
Dragon Quest stories have a cute exterior but tend to take a darker tone, whereas FF tends to have a more mature exterior are more lighthearted stories. Its gameplay, while mistakenly conservative builds upon each and every game in the series. While it's not trying out an endless tirade of new gameplay ideas, this means that DQ gets to work on perfecting a certain formula. FF's formula, while being perfected, has been dropped to become more contemporary and relevant each new entry. DQ has no such goals. What this means is that while you get one feature that's new to the series, the next game will likely have the same featured in a more evolved manner. Take the Alchemy Pot in DQVIII. It appeared in that game, but DQIX made it even better by making it instant and allowing you more customzation options tied to your vocation. Meanwhile FF had a fantastic idea in FFX-2's ATB battle system; it has since been left to dry. They made an amazing AI customization/strategy system in the gambit system that has gone to influence many other RPGs such as Dragon Age Origins, they've yet to bring it back and expand upon it. The result: when you play a DQ, you know what you're going to get. This is DQ's greatest asset, and also its curse. That said, despite accusations of being more conservative, DQ has been far more ambitious story-wise than FF.
The difference can even be seen in their names. Final Fantasy sound hopeful; Dragon Quest sounds like a journey - a journey of life with all the pits and falls that come that territory.
Both are great, but if I had to pick it'd be DQ and I grew up on FF (starting with VII as mentioned in the interview) and didn't play a DQ game until I was 20 years old (DQVIII). But he is right about nostalgia. I have far more nostalgia for FF than DQ and for that reason I have far more emotional attachment to the series.