I doubt anyone can actually give you a completely accurate answer without doing something along the lines of disassembling the game or being Edward McMillan, but after a rather successful run I think I can shed some light on this.
Chances of getting rare bosses: Boss enemies can be encountered not only on actual boss rooms, but also in seemingly ordinary room (this is especially common in The Womb). To me, it seemed that the frequency of special bosses is on par with the frequency of other special enemies, which in turn depends on the level you are on, as harder levels contain more rare enemies. When travelling in the womb, I'd have to say that 1 in four bosses I came across were rare, and less so on the easier level.
Do rare bosses drop anything special: Assuming an empirical answer can be provided for this, then after dispatching of at least 10 rare bosses, 3 of which in an actual boss room, I would have to say that the answer to this is no.
There is also the matter of Alt-Bosses, which are not a rare boss strictly speaking (they are more of a re-skin than a pallet change). These do drop specific special items as can be seen here and here.
Are there other differences to rare bosses: There are at least three that I am aware of, though I suspect there are many, many more. Here are a few examples from my latest run:
![Green Chub](https://i.stack.imgur.com/SJEHx.png)
This green colored Chub was actually slower than the regular Chub, and spawned boils instead of maggots. Be aware that the color doesn't necessarily indicate the rare boss's properties, as this page lists the version I described as black, and the version you described as green.
![Darkish Monstro](https://i.stack.imgur.com/iZ424.png)
I encountered this darker version of Monstro, which was slightly bigger than a regular Monstro. I can't say I noticed any additional differences. I also had a chance to encounter Monstros that are smaller than usual.
![Lime Larry Jr.](https://i.stack.imgur.com/KanvN.png)
This green/lime version of Larry Jr. actually fires projectiles at random, much like the Gusher enemy (and also makes a green mess of the floor).
Like I said, I suspect there are even more variations (some of which are chronicled in this wiki), but these are the ones I came across.
The Binding of Isaac wiki's got a pretty comprehensive (if slightly generalized, and minorly inaccurate) list. Remember that all cards are a single use item, so once they are used they vanish.
- O The Fool: Teleports the player to the starting room for that floor.
- I The Magician: Gives you the use of Spoon Bender (homing shots) until you leave the current room.
- II The High Priestess: Summon's Mom's foot stomp on a random enemy. Can also hit the player, and will do so if there are no enemies in the room.
- III The Empress: Gives you the use of Whore of Babylon until you leave the room. Whore of Babylon is Eve's curse, but unlike her version, this just hurts you half a heart then increases your damage.
- IV The Emperor: Teleports the player to the Boss Room.
- V The Hierophant: Spawns two soul hearts.
- VI The Lovers: Spawns two hearts.
- VII The Chariot: Acts as a single-shot use of My Little Unicorn.
- VIII Justice: Spawns bombs, hearts, coins, and keys around the player.
- IX The Hermit: Teleports the player to the Mini-boss Room, or the shop if a mini-boss room is not available.
- X Wheel of Fortune: Spawns a Slot Machine.
- XI Strength: Makes the player bigger, increasing maximum health and damage until the player leaves the room.
- XII The Hanged Man: Gives you the use of Transcendence (flight over gaps, rocks and spikes) until you leave the current room.
- XIII Death: Gives you a single use of The Necronomicon (massive damage to all enemies in the room).
- XIV The Tower: Gives you a single use of Anarchist's Cookbook (spawns multiple bombs around the room at random).
- XV The Devil: Gives you a single use of Book of Belial (increased tear damage) until you leave the current room.
- XVI Temperance: Spawns a Blood Donation Machine.
- XVII The Stars: Teleports the player to the Item Room, or acts as Telepills if there is no item room (i.e. in The Womb and Sheol).
- XVIII The Moon: Teleports the player to the Secret Room.
- XIX The Sun: Completely heals the player, does damage to all enemies on screen and acts as a Treasure Map and Compass for the current level only.
- XX Judgement: Spawns a beggar. If you have no money, spawns 1 heart, coin, bomb and key instead.
- XXI The World: Acts as a Treasure Map and a Compass for the current level only.
Tarot Cards II, III, XVII, and XIX were added in a patch on November 1 2011.
Best Answer
Those are part of a new item class called "Trinkets". They're like a mixture of useable items and passive items; the effect is always present, but you can only have one at a time. If you find another trinket and pick it up, you'll drop the one you have right now.