Dragon Warrior was the name adapted for the series when it when it was brought to the U.S., because there was a little-known tabletop RPG that already had the name Dragon Quest. In Japan, however, the series has always been called Dragon Quest. I'm not exactly sure why they decided to begin using "Quest" now, because it's only succeeded in causing a lot of confusion.
In short, Dragon Warrior 3 is the same exact game as Dragon Quest 3.
Another confusing factor is that the first three games were "remixed" on the SNES (their initial release was on the NES), and then again re-released on the Game Boy Color.
Here's some nifty trivia: Dragon Warrior 1 was the first console RPG released in the United States. In Japan, the very first console RPG was Fire Emblem.