First off, let me say that I wasn't 100% sure if these type of who would win debate posts were allowed. I didn't see any rule against it so I assume it is okay. Without further ado, the fight guidelines.

Talon Company from Fallout 3 VS The Regulators from  Fallout 3

Round 1: Standard Regulator vs Standard Talon Merc

Loadouts consist of what is considered standard gear for each (Merc uses combat armor and Chinese Assault Rifle, Regulator wears a duster and uses a combat shotgun)

Round 2: Squad of Regulators VS Squad of Talon Mercs

Round 3: Army of 200 Regulators VS Army of 200 Talon Mercs

Finally, because I am curious, the bonus round:

Which organization would be able to assassinate a New Vegas leader first (Being Mr. House, Caesar, or General Oliver)?
-Assassinating the leaders by whatever means realistically possible by either faction. Whether it be infiltrating, full-out assaulting, or nuking from orbit (somehow).

Please try and back-up claims with whatever evidence possible (I.E. statistics for weapons and armor and also lore based examples if there are any).

I think the Talon Company would win just because they have the better armor. I'm also sure the Talon Company has more members than the Regulators.

I'm with Yeezus, the Talons would likely work better together as a cohesive unit against the Regulators that are used to working independently. Having said that, I've been pondering rigging my game to be able to fight them both because in my storyline they would both be gunning for my character.

I think Talon Company would win because of their slightly better equipment.
However, I came here just to say that neither the Regulators nor Talon stand a chance against any of the mentioned Vegas factions.
Caesar and NCR have armies and armies, they would destroy both of these 'small factions' without doubt. And I really don't think I'd need to compare a securitron army (even more destructive if updated) to Talon/Regulators.
However this is the same as saying that they wouldn't win against BoS or Enclave.

Also, Caesar would massacre them due to the Legions nature.

Sorry should have been more clear about the last round, I'll edit it. The terms I was thinking of, and should have posted, was assassinating the leaders by whatever means realistically possible by either faction. Whether it be infiltrating, full-out assaulting, or nuking from orbit (somehow).

I really think Regulators and Talon Company are the same in strength since they basically carry the same weaponry. Although the Regulators have more vulnerable and lighter armor and seem to not have as many numbers. But really the reason why I see the two as equal is because it just depends who you'll get from your karma. But military wise, Talon Company seem more established and usually carry heavier weapons and travel with security bots. Hard to answer this since I usually only play as a good character in Fallout 3.

Talons are more organized, and also look cooler in my opinion ^__^

Well I'll answer that last question about assassination. I think Talon company would be able to. Ambush a small legionary squad. Take their armor and sneak into cottonwood cove and have a small scale assault. The only problem is they couldn't sneak in a large number of troopers. That question in general is a real tricky one. Assassinating Mr House would be a difficult task i think neither faction would be capable of. Kimball though, who knows, major public appearances in a time of war is quite stupid, so i feel as if both factions could pull that off.

Kimball would be possible for Talon. However, as far as Mr. House goes, he just has to close the Lucky 38's doors and its impossible they get in.
And finally, Caesar, also impossible, because yes, sure a small group of Talon could infiltrate Cottonwood Cove, and lets say they are able to sneak to Caesar's Tent (I highly doubt this), they don't stand a chance against all the pretorians inside. Also, we know how it went for Benny, sneaking into the fort.