Final Fantasy, like Dragon Quest I and II has a very clear contained 'starting area'. Dragon Quest I has almost the entire world open to you at the start, but enemy power levels definitely keep the player contained into a smallish area of the starting town and the cave to the north. The game starts with the king giving you an introductory talk. The cave to the north is a fairly simple dungeon - something like a 14 by 14 square, with the entrance in the upper left, and a chest in the lower right. Said chest gives you your goals in the game. You technically don't need to check said chest or dungeon to advance, but it's a good way to gain levels and explore early on. Dragon Quest I, particularly the NES version, feels far more simplistic and archaic in plot and gameplay systems.
In Dragon Quest II, you get an introductory cut scene, and a knight carries the news of evil beginning to rise to the next town over. Your hero is told by the king of his mission, and you head to find your three partners. After going to a few different towns (and covering far more ground), you can access the Cave of Heroism, and are told that the prince visited, and was heading home. Catching up with him in town, you gain your first party member, and unlock a number of areas. One notable thing in Dragon Quest II is that it's almost impossible to do certain areas without gaining a specific level (for example, you need to be about level 8 to do the Cave of Heroism) and that there is a great deal more land involved in the game. Your starting area is a quite large location.
In Final Fantasy, you have the famous princess rescue. You start the game outside the starting town, head in, hear about the king's plight, and begin your mission. You head north, again, and enter a spooky dungeon full of bats. The edges of the dungeon tend to have stronger enemies, which is a nifty way of making a place feel unwelcoming. After a very easy boss, the princess is rescued, and there's a mini-cut scene as the bridge is repaired. It's quite noticeable that the early game in Final Fantasy is D&D esque. You have the spell system with number of casting of magic at various levels. You have D&D esque monsters / races, with an implied assumption that these are normal (for example, you face a Dark Elf and find dwarves and elves very early in the game.) There's even a 'turn undead' esque spell.
Music-wise, Dragon Quest I actually is more interesting, to my ears, than II. Part of it is probably due to the iconic nature of some of the sound effects / musical stings that are used again and again in the series. However, it is striking that the world map music and the battle music are very smoothly integrated. After the "winning the battle" fanfare, the world map music / wherever you are music kicks in again, resulting in a smooth transition from fighting to travelling. Dragon Quest II does not overlay the world map battles over the world map, and there is a stronger difference in tempo from the world map music (which varies over the game,) and the battle music, so there is less of a smooth transition. On the other hand, the music is definitely more complex in scoring. Final Fantasy's music has been remixed and varied over so many games in the series that it simply sounds familiar to my ears. Much like Dragon Quest, much of the music is catchy and addictive, though the map to battle transition is far closer to II in that there's a fairly strong musical sting to warn you.
Comparing the three games, Dragon Quest II really seems much closer to Final Fantasy. You have multiple party members in II, much like your 4 characters you choose at the beginning of Final Fantasy. You have a fairly vast world in II (the world map of I is about 1/4 of II's world map) much like the 4 large continents in Final Fantasy. Final Fantasy, particularly the original NES game, has some pretty big areas of nothing - a good early example would be checking out the distance and the path from the first town's bridge to Matoya's cave. Dragon Quest II has similar issues. A good example would be to look at the path from (using Japanese town names, since that's what the gamefaqs world map uses) Moonpeta, past Moonbrooke, down to the shrine, up to the Dragon Horns, and to Rupugana. The early game starts you in Lorasia, using the Japanese names again. Note the amount of land between Lorasia and the early locations.
Gameplay wise, all the games in the NES incarnations need a lot of levelling to progress. Dragon Quest I is a little more unforgiving, since you have only one party member, and lucky shots on other party members cannot help you squeak by. Having to choose to climb stairs, and having the search command tell you "There's a treasure chest here" without opening it varies between being a quirky limitation and infuriating. Since sleep and stopspell have a chance of failure, certain bosses can be almost unbeatable if the randomizer hates you and you are low level.
Dragon Quest II, gameplay wise, is clunky. You have a limited inventory that quickly fills with keys to open various types of locked doors and quest items, leaving you struggling to throw out various battle spoils to gain room for items in the late game. Your party has figured out how to use stairs, and most dungeons either have a ledge to hop out or a quick escape item. Ships can only be summoned via saving and resetting in a location, or sailing to that location. More towns have places you can save your game, which is a nice feature. Certain enemies are vary unbalanced, and can cause a swift death to the unwary.
Final Fantasy has no way to 'warp' your ship to you, but you never really need it after the airship is unlocked. Dragon Quest I and II have keys that unlock special things in the early game areas. Final Fantasy (NES version) gives no reason to return. There is a certain pleasure in seeing enemies that used to destroy you running in fear or dying easily. There is an ingame map, which is handy.
It's also pretty clear why some people find Dragon Quest II to be a flawed game. Your party members are forced to have a level difference between each member, since you must hit a certain level to easily obtain said party member. You have some almost ridiculous treks - take that Moonpeta to Rupugana walk. Armor and weapons, for the most part, are very expensive (in the NES version) and largely useless. There's very few upgrades for certain party members. The dungeons, if you know the routes, are very quick to complete, and if you go off the correct path, you frequently get useless rewards (this is changed in remakes.) A particularly bad example is a dungeon with a quest item on floor 3, and 4 more levels to find a chest at the very top - containing nothing. In a remake, it's a very nice defensive item.
Talking about rewards, Final Fantasy actually assumes you don't explore every dungeon completely. Certain chests 'open' when you open one on the other paths, resulting in you being unable to get the contents twice. You're actually encouraged to complete the early game as fast as possible, since a lower level when you hit a certain event will result in better stat growth. There is one annoying looping maze, which in the GBA remake, is all the more annoying, since it's hard to tell if you found an up stair or a down stair when you find one.
Dragon Quest I has a very gear and level based progression in the game. You can search dungeons in depth to gain a few prizes, but mostly, dungeons exist to gain a destination, or to obtain something. Most dungeons in the game feel a lot like someone scribbled simple walls and damage floors on graph paper. In Dragon Quest II, there's a much more 'natural' feel to caves and towers, making it seem closer to Final Fantasy in nature.
Travelling also varies. Dragon Quest I has you always going back to the king in the first castle to save (and get free healing.) Everything other than the return to king spell is done on foot. For the most part, you can gain access to areas fairly quickly, and you almost always have a base of operations semi-nearby for healing until the very late game. Often, you have to stay near a town and fight monsters to gain enough levels for the next area. II has a ship (obtained with almost no fanfare via rescuing a girl.) The ship can only be teleported to your location via saving and restarting the game. Warping does not bring the ship with you, but you can teleport from town to town with more control. Final Fantasy has a canoe, a ship, and an airship, though in the late game, you really only use the airship. The vessels are treated with a bit more fanfare. There is no way to teleport from town to town.
So - summing up what I've noticed, Dragon Quest I is a very early RPG, but the production values make for compelling addicting gameplay. The music flows smoothly from track to track, and the monsters and backgrounds have a noticeable Toriyama charm to them. It's a very slow game, more interested in building up to be able to explore rather than monster smashing and the things you explore are mostly not that interesting. Still, it definitely is still playable today, with patience. Dragon Quest II feels like an ambitious but highly flawed game. Since it came out around when Final Fantasy did, it's very easy to compare them. Unfortunately, the act of playing II often sours whatever pleasure you can take in the storytelling. The slow speed in II feels like it's slow for the sake of slowing down gameplay and to show off the world. Exploring for the sake of exploration is interesting, but you gain very little in 'value' from it, save for levels. Final Fantasy has a rudimentary plot, and a strange mix of Dragon Quest like ideas and D&D like concepts. The scope of the game environments and the various choices for ease of gameplay do make the game stand out in comparison to Dragon Quest II. The comparison does explain how Square hit lightning with that game. The early game is a bit too hard for my taste, but it's not hard to see why people enjoyed the game.