I can't seem to find a good listing of the differences online, so I'll try to compile a short list. If you're interested in my personal opinion, I discussed my feelings about the major changes here. (warning: personal blog link) The first thing to note is that Modern Warfare 3 follows Modern Warfare 2 pretty closely, and isn't as related to Black Ops. There are a lot of similarities, however.
There's a lot of commonality between each year's installment of Call of Duty when it comes to perks. You've almost always got the "silent footsteps" perk, the "invisible to UAV/air support" perk, the "fast reloads" perk, etc. There's a listing of the perks on this page, which goes over the perks available. You have 3 perk slots per class, and much choose a perk from each tier, like both Black Ops and Modern Warfare 2.
Weapons unlock based on your level, which goes up based on your XP, which you gain by completing matches, objectives within matches, kills, pointstreaks, bonuses, etc. There are no point costs for unlocks; whenever you get to a particular level you'll unlock all the items from that level.
Weapons also gain XP, which is measured separately. Each weapon has a separate XP meter, and the higher your weapon level, the more attachments and camo options you'll have available. You can also select a "weapon proficiency" which is similar to a perk, but attached to your weapon. In previous iterations of the franchise, one of your 3 main perks would usually allow you to equip 2 attachments on a weapon. In this game, this is controlled by a weapon perk. However, most of the weapon perks are new, and grant bonuses like increased damage, range, or reduced recoil. Attachments are mostly the same, such as sights, undermounted grenade launchers, front mounted grips, etc.
The available weapons are quite similar - assault rifles, SMGs, LMGs, sniper rifles, and shotguns all return, and most are pretty similar, especially compared to MW2.
In previous iterations, you got to choose a set of 3 bonuses that unlocked as you got a number of consecutive kills. The developers this time around decided to reduce the focus on kills, and try to reward a variety of contributions made by players. Shooting down air vehicles, capturing objectives, and gaining assists (if you have Hardline Pro as a perk) all earn you points towards your pointstreaks, among other things.
There's also the concept of "Strike Packages." There are three available, Support, Assault, and Specialist. Assault is the "classic" strike package, where your pointstreaks unlock as you gain successive points without dying. Support changes this up by allowing you to keep your pointstreak even if you die, but the rewards are more focused on bonuses to your team instead of dealing direct damage to your enemies. Specialist trades your normal pointstreaks for the chance to gain extra perks as you get successive points without dying, up to the point where you have all the perks on your class at the same time.
Most of the pointstreaks are things we've seen before, such as UAVs, air strikes, chopper support, sentry guns, etc. Care packages return, as well as the option to take a Juggernaut armor care package that makes you far more resistent to damage, but less mobile.
Where you usually could take a tactical grenade, a lethal grenade, and a piece of equipment (ie, radar jammer, tac insert, etc) you now get one piece of lethal equipment (including grenades or 2 types of mine) and one piece of tactical equipment (such as a portable radar or flash/stun grenades)
Secondary weapons are more in line with Black Ops, where your selection is limited to pistols, machine pistols, launchers, etc.
In previous games, prestige mode meant that you'd lose all of your unlocks and have to start over from level 1, as if you'd just bought the game. In Modern Warfare 3, you gain "prestige tokens" from successive prestiges, which allow you to (among other things) unlock gear from level one. You will have to level the gear up as if you just unlocked it, but you won't have to wait until you get back to your old XP levels in order to use your favorite weapons or perks.
Most of the multiplayer game types return from the previous games, although a couple of new modes have become popular. For instance, Kill Confirmed is very popular. It is like Team Deathmatch, but players drop tags that must be collected in order to get credit for the kill. This mode emphasizes close-range combat over long-range engagement.
The multiplayer maps feel smaller, and more of them are the size of Nuketown or Rust than in previous games. There are fewer Array or Wasteland-sized maps, for instance. It's much easier to roll a short-range class and dominate than it was previously. Shotguns and SMGs are much more frequent in classes than sniper rifles, although they still have their place.