There is no disputing the hardware differences between Microsoft's Xbox One, and Sony's PlayStation 4. On both sides we have 8GB of RAM, 500GB of hard disk storage space and similar GPU architectures. Despite the fact that both systems are utilizing AMD technology in their chips, the PS4 outperforms the Xbox One.
The PlayStation 4 comes with a Radeon-based GPU with 1152 cores clocked at 800MHz. The Xbox One also includes a Radeon-based graphics processor but with only 768 shader cores clocked at 853MHz. Even though the Xbox One tries to trounce its shortcomings by using 32MB of SRAM as a buffer, it still does not perform as well as Sony's newest gaming console. Due to this fact, Sony is claiming that "Watch Dogs" will run best on the PlayStation 4.
PlayStation's "Watch Dogs" page claims the PS4 will be the only system to run the title in full 1080p at 60FPS, although this has yet to be confirmed by UbiSoft. This claim may have been made due to the Xbox One suffering a loss in resolution in games such as "Battlefield 4" compared the PlayStation 4 version.
It is still too early to determine which system will run the game best due to fact that many developers are still transitioning to the new hardware, however it is certain that "Watch Dogs" will create a huge fan base on both the Xbox One as well as the PlayStation 4 regardless of the resolution.
Update: Sony has recently taken down these claims on PlayStation's "Watch Dogs" website. Sony may be attempting to cease the information from spreading, or the settings that the title runs at is still to be determined. It is unconfirmed if the game will run at 1080p at 60 frames per second.