I've seen a lot of complaints on other android related sites about the One XL having "only" 2 S4 Krait cores rather than the One X's Tegra 3 quadcore. However, I'm personally glad that's the case. I'd probably buy the LTE version even if I didn't have LTE coming to my area for the entire time I would have the phone. Here's why.
Performance
But, but, but, 4 cores are more than 2! True, however there are a lot of times all four cores won't be used. Android's UI is single threaded (called the UI thread). That means all scrolling, highlights as you press a button, toast notifications, notification bar notifications, swipes, transitions, everything you see basically happens on this one UI thread. If you want a little more information, look here.
In that case, it makes sense that you would want a processor that is powerfull when it comes to single threaded tasks, right? As it so happens, the difference between the Tegra 3 and the S4 in single threaded applications seems significant. Heck, it's a huge difference from the previous champ the SGS2:
That should translate to some incredibly smooth UI. In fact, Anandtech stated in their preview that the S4 was the fastest they've ever seen ICS run on anything before.
However, let's say you are running an app that has spawned a ton of worker threads (for some reason) or are doing multiple things at once that use all the cores. Surely the Tegra 3 will catch up with it's four cores. Right?
Maybe not. This S4 is impressive. I don't expect anywhere NEAR this big of a difference in real world use, but I do expect to see an S4 advantage.
Battery
This will be very interesting when reviews of the One X and the One XL come out. I'm not sure who will win. The Tegra 3 has the companion core, where they S4 has LTE. However, it's the first chip with integrated LTE (much better for battery), and it is also done on the 28nm process rather than the 40nm process on the Tegra 3. Basically, it's more power efficient that way. So it's companion core vs lower power cores, and LTE with integrated LTE modem rather than a separate, powersucking radio.
Games
The Tegra 3 is better at games, sometimes. They seem to trade blows fairly evenly. However, GPU's where most of the gaming bottleneck will come in, so more cores may not help here. However, both chips soundly beat out devices like the Galaxy Nexus, the Razr, the SGS2, etc. How many games do you know that won't play on those phones? Both chips will be excellent for gaming, and both chips will have their own game stores (sadly). However, the Tegra Zone is already established with a good sized library. Hopefully Chainfire 3D will let you sort this out.
In all, these are both great chips, and the phone they are inside is going to be awesome. However, don't hate on the dual core
Sources:
AnandTech - Qualcomm Snapdragon S4 (Krait) Performance Preview - 1.5 GHz MSM8960 MDP and Adreno 225 Benchmarks
AnandTech - The Qualcomm Snapdragon S4 (Krait) Preview Part II
Performance
But, but, but, 4 cores are more than 2! True, however there are a lot of times all four cores won't be used. Android's UI is single threaded (called the UI thread). That means all scrolling, highlights as you press a button, toast notifications, notification bar notifications, swipes, transitions, everything you see basically happens on this one UI thread. If you want a little more information, look here.
In that case, it makes sense that you would want a processor that is powerfull when it comes to single threaded tasks, right? As it so happens, the difference between the Tegra 3 and the S4 in single threaded applications seems significant. Heck, it's a huge difference from the previous champ the SGS2:

That should translate to some incredibly smooth UI. In fact, Anandtech stated in their preview that the S4 was the fastest they've ever seen ICS run on anything before.
However, let's say you are running an app that has spawned a ton of worker threads (for some reason) or are doing multiple things at once that use all the cores. Surely the Tegra 3 will catch up with it's four cores. Right?

Maybe not. This S4 is impressive. I don't expect anywhere NEAR this big of a difference in real world use, but I do expect to see an S4 advantage.
Battery
This will be very interesting when reviews of the One X and the One XL come out. I'm not sure who will win. The Tegra 3 has the companion core, where they S4 has LTE. However, it's the first chip with integrated LTE (much better for battery), and it is also done on the 28nm process rather than the 40nm process on the Tegra 3. Basically, it's more power efficient that way. So it's companion core vs lower power cores, and LTE with integrated LTE modem rather than a separate, powersucking radio.
Games
The Tegra 3 is better at games, sometimes. They seem to trade blows fairly evenly. However, GPU's where most of the gaming bottleneck will come in, so more cores may not help here. However, both chips soundly beat out devices like the Galaxy Nexus, the Razr, the SGS2, etc. How many games do you know that won't play on those phones? Both chips will be excellent for gaming, and both chips will have their own game stores (sadly). However, the Tegra Zone is already established with a good sized library. Hopefully Chainfire 3D will let you sort this out.
In all, these are both great chips, and the phone they are inside is going to be awesome. However, don't hate on the dual core

Sources:
AnandTech - Qualcomm Snapdragon S4 (Krait) Performance Preview - 1.5 GHz MSM8960 MDP and Adreno 225 Benchmarks
AnandTech - The Qualcomm Snapdragon S4 (Krait) Preview Part II