My gut reaction has always been to say that Mega Man 3 is the easier of the two, but that's not entirely true. I had the opportunity to watch an inexperienced player attempt both games recently, and a more accurate statement would be that Mega Man 3 is more stable. MM2 has difficulty spikes that stand out in one's memory, such as Heat Man's blocks, Quick Man's lasers, and BooBeam.
However, most of the main stages of MM2 can be beaten with less hassle. Crash Man and Metal Man are just speedbumps, while Bubble and Air front-load most of their difficulty. Flash is full of Sniper Armors, but offers players a safe route around them. That leaves Quick and Heat as the stages a player is likely to get stuck at, while Wood Man is a very difficult boss without the right weapons.
Then we have Mega Man 3. Nothing in the first eight stages, or anywhere in the game for that matter, rivals the worst of MM2. However, almost every stage has something toward the end that can cause a few deaths. Snake has both the clouds with dodgy hit detection and enemies trying to knock you off of small platforms. Shadow has the Parasyu section while Hard wears the player down before throwing them into the toughest fight with Proto Man, and both have a difficult boss.
Gemini and Spark both end with the hardest platforming segments in the game. That leaves Magnet, Top, and Needle as the easiest stages to start with, but Needle Man is among the hardest bosses, and even an easy vanishing block section can get players stuck for a while. Top is probably the best starting stage, but that depends on how quickly a player can learn to deal with the spinning platforms at the end.
Needle Man also guards the Rush Jet, which trivializes the game's platforming sections, but a player would have to beat at least Gemini and probably start with Snake to get it if they can't handle Needle Man with the Mega Buster. Point is, a new player could try and fail most of these stages before finding one they can beat, whereas Mega Man 2 has at least four that are more forgiving to players without the right item or boss weakness.
That said, Mega Man 2 keeps throwing dangerous things at the player throughout the Wily stages, as well as making Crash Man's lifts dangerous and allowing the weapons to be used for puzzle solving, while MM3's Doc Robot stages are easier than their counterparts overall (though having two boss fights in each helps balance that), and the stage difficulty falls apart entirely in Wily's castle, leaving it up to the 2nd and 3rd bosses to pick up the slack.
Difficulty aside, Mega Man 3's major changes from 2 include the slide and the upgrade from Items to Rush. I like the slide overall as it gives Mega Man an alternate way to dodge things and avoid bosses, as well as providing an easy method for one-way exits on split paths, but it also adds the possibility of a player doing it by accident and sliding into a pit. How much of a problem that is just depends on how often that happens to a particular player. For me, it's almost never an issue.
I think Rush Coil is mostly a miss. It only had a couple moments that justified its use as anything more than a tool to overcome obstacles that only exists as reasons to use the tool. Starting with it also removed a lot of the fun in getting optional items. Mega Man 2 hid them on alternate paths and in places that required an Item to get to, so they were more like rewards for having beaten certain bosses early, but having Coil from the beginning means that getting out-of-reach items is just a matter of whether the player wants to bother opening the menu and using it.
Rush Jet is incredibly broken, allowing Mega Man to fly over any platforming segment in the game once he has it, but it's held by a fairly strong boss, it opened up the ability to add flying segments, and it added a completely different way to deal with certain bosses. In that sense, I would call it broken in the same way as the Metal Blade. That is, a fun kind of broken that makes players excited to have it. Rush Marine only got a handful of screens to be useful in, but it was fun for those segments and certainly didn't hurt anything by being there.
The weapons somewhat reflect the overall feeling the games leave behind. Both games had some bad weapons, or at least weapons that weren't given a chance to shine. However, the good weapons in Mega Man 2 were really good, with Metal Blades and Quick Boomerangs tearing up most enemies, Leaf Shield to protect Mega Man from things his Buster has a hard time dealing with, and Atomic Fire to blast away the largest enemies.
Mega Man 3's Buster replacements were the Needle Cannon and Shadow Blade, the former of which was essentially an autofire button and the latter more of an alternative than an upgrade. The rest were powerful in the right situations, but it was more difficult to figure out what those situations were. That goes extra for Top Spin, since a player has to risk taking a hit every time he tries it out on something new. Then again, it was a lot of fun to use and the game gave players a big reward for doing it, with two Wily bosses destroyed by it outright, including the very last. MM3 also had fewer opportunities to do neat things with them during the stages in the latter half, opting to put a stronger focus on Rush instead. They were still helpful, they just didn't have as many moments where a player could completely stomp something with a weapon that gave them trouble before.
It's a tough call, but I'm going to say that Mega Man 3 is my favorite of the two. Its greatest moments are less flashy than Mega Man 2's, but the weaker sections are also spread thinner and there's a constant stream of new stage objects, with each main stage being fun in its own way. The Wily stages were handled much better in MM2, but I think the Doc Robot stages help even it out, and MM3 can be forgiven for rushing the end after putting players through that.
However, it also makes sense to hold up MM2 as the gold standard. MM3 added a lot of good things, but it also made a major addition to the controls and took the focus away from weapons somewhat. The overall design of the series started to change here, while MM2 was a more direct upgrade to the first game. It makes sense for someone making a new game in the series, or a similar type of game, to first look at what Mega Man 2 got right and work from there. Both are quality games, and both deserve their spots as fan favorites.