Lets say they both woke up in a new world and had one goal to take over the world. Who will win?
Sorry for the lack of detail.

I assume you mean Lelouch and Light?

I'd probably say Lelouch. Killing people with a death note is great. But Lelouch's power is more flexible in taking over the world. And his battle tactics are pretty amazing. He looks long term, at all times.

I'd have to side with Lelouche as well.

Light has the more powerful weapon in my opinion in terms of destructive power.
But it can only go so far in getting others to follow him, may it be out of fear or religious worship.
Lelouche can mind control it, or simply gain sincere loyalty from his tactics and ability to lead.
Plus Lelouche always keep a clear and focused head on his goal, which Light became obsessed with playing and being God.

If they were to face each other though? It all depends on how well Lelouche can hide his identity like L did (Which he did right up to around the end).
If he can pull that he'll probably turn the tables on Light. Otherwise Light get's an easy victory.

What review what we know, Light's plan backfired which pretty much got him kill. Lelouch pretty much took over the world before planing out his own death like a G.

Light<Lelouch

Light would never have been capable of 'taking over the world' even when he didn't share a reality with Lelouche, his goal was to usher in a new era for the world in which Kira was the god. There's a difference between people fearing you as a faceless god of death and directly ruling over the world. The death note isn't really an effective tool for straight up taking over the world, as you'd be ruling through fear-tactics with no loyalty and the only thing protecting you being that people don't know who you are (And Light already proved in the show that he couldn't keep his identity secret forever).

So yeah, Lelouche definitely. Though it would also be a likely possibility that it would end in a tie, with Light killing Lelouche but ultimately failing to take over the world and being killed by the people.

LeLouche would win simply because he'd identify the greatest threats in the Death note universe and then use his identity as Zero to eliminate Light & company. After all, the Death Note only works if it has your name, and LeLouche is shown as repeatedly hiding that detail from the vast majority of people... So...

Well, that, and Light has a notebook. LeLouche has a small army of dedicated rebels in mech suits.

This is literally the epitome of the US Military versus [insert nation in Middle East here].

Lelouche, hands down. For reasons that already stated above and he hid his identity behind mask. No one knew his real identity, except for a very select few.

-> Lelouch's power could work once for every person. How many people in the world again?
-> Light could write names in the Death Note to kill them. Now this is something I've always wondered, could the Death Note ever run out of blank paper to write on? >3 Well, if they could, then Light is screwed when he run out of places to write names in his Death Note.

And if ever Lelouch and Light meet directly face to face? Lelouch could zap Light's will through eye contact in a split second before Light could find out his name, much less writing it down in his Death Note. x3