Starting with Dragon Quest III, one can see that there's an immediate difference between Dragon Quest II and III. While you're summoned to a castle in the first few minutes of the game, you do not start in a castle. You start at a very low level again, but you can instantly pick up a full party (also very weak.) You can create a custom party even via choosing a man or a woman from a set of different jobs. Much later in the game, you can change jobs to modify how characters advance. This is a sizable difference compared to Dragon Quest II, which had you picking up a warrior mage, and an incredibly fragile mage as you progressed. The default party is the hero, a warrior, a healer, and a wizard. There's also a day night system, and a choice as to the gender of your hero.
Final Fantasy II also does not start with going to a castle. You have the scrolling text intro, much like Final Fantasy, and you're thrown instantly into a battle that you cannot win. You proceed to lose a party member after the battle, and your home town is a cluster of rebels hiding with a dying king. Final Fantasy had evil in the dungeons lashing out at the towns, but there was not a clear and present danger to the town in the early game. You change your party member throughout the game (plot determined) and the world changes as the plot advances. So, you can see that there is a distinct change from the previous games.
Let's start with the battle system. Dragon Quest III is pretty much Dragon Quest II's battle system with another party member. Each job class has a different equipment set, and each class has strengths and weaknesses. Strategy wise, there are some classes that are very weak, but very useful when fully levelled up. Women seem oddly weighted in the game. Your hero of the game can be a woman, though in the English translation, the game seems to think she's rather convincingly cross-dressed due to the occasional gendered term errors. Women seem to get much better statistical options, and remakes of the game continue this trend. Compared to Dragon Quest II, the game does 'progress' more quickly. You cannot do the first dungeon in II until you're near level 8. You can do the first dungeon in III at about level 5. There's also a fair amount of things designed to make it easier to explore. Fairly early in the game, you gain warp spells to go back to a town and items that do the same thing. The first dungeon has several ways to let you escape quickly, and an inn within it.
Final Fantasy II's battle system is infamous. The creator went on to make the Saga series of games which tend to showcase amazing music and a steadfast unwillingness to be accessible. The most infamous of these is Unlimited Saga, which amounts to a roulette game for almost every action, from attacking to progressing in a dungeon. At first glance, Final Fantasy II looks conventional. You have magic. You have fighting. You have items. You have no experience values though, and no levels.
Instead, you gain stats depending on what you do in battle. This is supposed to work via having you do poorly at something, or do it a lot, and you gain stats or skill in doing the thing. Unfortunately, there's a bug in the NES version. You see, to level up strength and the level of your skill with a weapon, you're supposed to select the weapon and fight. However, if you pick the attack, and cancel out of the attack, it counts as one attack. Ditto for casting magic (though that does not level up your magic spell, just your magic skill.) Do 100 action and cancels and you gain a level. You cannot do this with the last character, but during most of the game, this is a guest character. To gain HP, the best method is to either use a spell to swap HP with an enemy, or to hit your party members. In the NES game, it's not really possible to play the game without messing with the system deliberately. If you over level, you may find yourself unable to finish the game.
Your three permanent characters are the usual archetypes from Dragon Quest II. Your hero is a generic character. You have a female character (low HP, high magic potential.) You have your big bruiser of a guy who has low magic potential and high HP and attack. The guest characters vary from a female pirate, a mage, a cowardly young man, a monk, and so on. You can, due to the system, make anyone into the type of fighter you want. Women characters are all punished strength wise. The encounter rate is pretty high and status effects are very debilitating (high chance of success and they do not let you gain stats after a battle.) You can quickly leave a dungeon, but HP drops to critical when the person casts the magic. Due to the fact that magic levels up (and gains higher costs), saving magic and using items is one way to control the level of your magic. This means that you never feel that capable of handling a dungeon, since you can't just heal your party via magic.
Graphically, Dragon Quest III is interesting because there's a pretty obvious attempt to make areas look different. For example, the first dungeon can be entered from two points. One is a sort of a basement in a hut in the woods, which lets you into a castle like area (with a locked door.) The other entrance is a maze like cave that lets you into the same castle like area. Up from that, you find a green walled castle, and up from that is a short tower, decorated in magenta and statues. This makes it easier to tell where you are and gives variety and 'naturalness' to the landscape. Due to varied layouts in some dungeons, you can tell what floor you're on fairly easily.
Final Fantasy II's dungeons have varied layouts from floor to floor, but they are pretty maze like. In Final Fantasy, there were chests that'd open once you checked another chest to discourage exhaustive exploring. There were also 'spiked floors' which cause a battle when you approach a spot. In Final Fantasy II, many chests have a "monster in a box." These tend to be difficult battles, sometimes harder than the actual boss of the dungeon. The rewards are not often worth the trouble to fight the enemies. It is possible to finish some dungeons without checking all paths.
Unfortunately, a more in depth analysis is hampered due to Final Fantasy II being almost unplayable in the NES form. The line between success and death in the game is very thin. The GBA and PS versions are more generous, but it's still a very quirky and difficult game. For example in the Dragoons quest arc, you have to enter a dungeon and put an egg in a spring. The boss of the area is a chimera with about 700 HP. There's a random enemy with about 750 HP. The random enemy hits for about 100 - 300 HP per hit, and at this time in the game, you have between 900 to 500 HP and a new very low HP party member is usually the low end of the scale. The random enemy is also very hard to hit, averaging about 80 HP per round, unless if you optimize your weapon use for him. The chimera can be hit for about 200 HP per round, assuming a less than optimal equipment layout. This is not balanced.
Dragon Quest III on the other hand, is quite playable, though almost aggressively grinding due to the size of some of the later dungeons (the tunnel from the first area to the second is particularly annoying). So far, I'm to the king with the missing crown in Dragon Quest III, and I'm to the Collesium in Final Fantasy II. Perhaps at some later date, I can say more.