How to play as Judas

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:

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.

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.

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.

According to comments on the wiki, "Everything is Terrible" makes alternate colored monsters and bosses more common. This is actually good, because alternate colored monsters tend to leave drops on kill.

According to the mechanics guide, it also makes Curses more common in Wrath of the Lamb.

Additionally, however, bosses drop one less heart pickup (heart, half-heart, spirit heart) than they previously did.