Blade of the Ruined King Vs. The Bloodthirster

Initial Thoughts

It seems like BT is the best choice if you look at it fully charged. But I'm not sure. In terms of lifesteal - healing the BotRK returns half of the damage from its passive back.

Which means (very simplistically) that to a champion with 2k health (which is pretty medium) and NOT taking into consideration the armor (I will do the same for BT thats why I say simplistically) it deals 80 damage and you heal for 40.

The extra lifesteal of a FULLY charged BT is 8%. So that being said IF you have 400 damage (which is a lot) and the opponent has no armor you will deal 400 damage and heal for 32.

So the way I see it in terms of lifesteal, BT might actually be worse. It's also definitely more expensive and its also a kind of snowball item so its risky. You will most likely not have it fully charged the whole game.

In terms of actual damage for the same opponent of 2k health and no armor (lol)
You will get a bonus 70 damage per hit with BT and a bonus of 80 damage per hit with the Ruined King. Of course the lower health the opponent the lower the effect of Ruined King.

The only aspect that gives BloodThirster an Edge (in my opinion) is critical damage and skills that scale with AD.

When you crit that extra 70 damage I mentioned above becomes 140. The on hit effect damage from Ruined King can't crit. So if you have an Infinity's edge and a phantom dancer or Trinity Force maybe late game Bloodthirster will definitely give you more damage. (Unless you got a mundo with 4k health there.)

As far as skills are concerned it's pretty straightforward that BT gives you more flat AD so it boosts your spells more than Ruined King.

And don't forget you get the active from Ruined King too.

Final Thoughts

I totally forgot the fact that its nothing like Madred's. It uses CURRENT health not maximum. So as you can see if you have a champion with current health less than 1500. You deal less damage than BT and I'm pretty sure you're gonna heal for less as well.

That being said in combination with everything I said above, I'd go with Ruined King only with opponents that have a maximum health of over 3k.

Outside Link: Here you can find specific numbers which pretty much draw the picture of this duel

Post Patch 3.03 Thoughts

They now changed the item as follows:

Stats    
+25 attack damage
+15% life steal
+40% attack speed
Passive  Unique: 
Your attacks deal 5% of the target's current health 
in physical damage (60 max vs minions and monsters).
Active   Unique: 
Drains target champion, dealing 15% of the champion's maximum 
Health in physical damage and healing you by the same amount. 
Additionally you steal 30% of their movement speed for 4 seconds 
- 60 second cooldown. (500 range)

I like these changes a lot mainly because it no longer gets shadowed by Bloodthirster. Its now an item for a completely different purpose. If you want attack damage with your lifesteal the choice is even clearer now - Bloodthirster. But if you want the extra attack speed and an (even better than before) on hit effect against health tanks Blade of ruined King is your clear choice. By making this item more unique I consider this a buff.

As a sidenote: After they took out Mardred's On-Hit-Effect Tanky Teemo (Malady, Wits End, Frozen Mallet, Mardred's) was seriously hit. With this item this build is back in town (not as good as it used to be with Mardred's but still its nice to have the option to play it)

Which provides more lifesteal, the only variable being the stats of the weapon?

This is impossible to answer, since BotRK does damage based on your opponents health. Genrally, BotRK will heal more when attacking champions because of their high health but BT is better for sustain from minions or jungle mobs.

What is better for what situations, and when should you build BT instead of BotRK and Vice Versa.

Because BT provides more AD (70-100 vs 40), it's going to perform best on champions with skills based on AD. Consider Riven, who's every ability scales on AD. The extra 60 AD she can get from BT is amplified by each ability, making it a clearly superior choice.

Alternatively, champions who benefit from attack speed will do more damage with BotrK's passive. Teemo is a good example of an attack speed champion who's skills don't rely on AD.

BotRK provides an active that deals damage, heals your champion, and slows your opponent. This is a great ability in general, and is especially useful on champions without slows/stuns. BotRK will also deal more damage to high health targets, and combos well with Last Whisper to shred through tanks.

tl;dr Pick BloodThirster for champions with AD based skills and Blade of the Ruined King for attack speed or on-hit champions. BotRK is also very good at shredding tanks.