Contrary to popular belief, the LS wasn't just another Jaguar. The LS was a much larger car than the S-Type and used a different V8. Unlike the Cadillac Catera, this vastly improved competitor proved a success in hometown rival Lincoln's career into building an "entry-level" luxury car for the masses and especially for the younger audience, when the Continental and Town Car were geared toward senior citizens.
The LS Sport was the better variant of this car. Sure it had the same power and torque and everything, but in addition to those, it came with improved wheels, tires, brakes, suspension, and a paddle-shift transmission to target younger buyers.
This was one of the rare moments when Cadillac had to compete with Lincoln and not the other way around. The CTS-V came with 400 horsepower, but at the expense of displacing a whopping 1.8 or 2.1 liters more (depending on model year).
Both the CTS-V and the LS V8 Sport were athletic machines on the track. However, the LS V8 Sport was a beast since it wasn't trying to be what it wasn't — the glorified, overpowered wannabe "muscle cars" that the CTS-V (and later rivals like the Chrysler 300C and 300 SRT8 models) were.
Sporting 104 cubic feet of passenger volume, the LS was a true full-size sedan (equal in size to the Seville, barely smaller than the Oldsmobile Aurora and Chrysler 300M, and slightly larger than the Buick Regal); and while at 99 cubic feet, the CTS was only a midsize (luckily for the CTS, it was larger than the 9-5, E-Class, S-Type, and A6; and equal to the 5 Series and S80; but it was still refusing to stick to the American tradition of the full-size class).
Whereas the CTS was partly an evolution of the Catera (the Sigma platform was really an improved V-Body), the LS was something different. The DEW98 platform was truly American, even if the Jaguar stablemate was British; it unveiled the retro 2005 Mustang (on a different platform name, that is D2C); in comparison, GM knew that the Sigma platform was highly outdated and needed to rely on the Zeta platform instead (used in Holden Commodore and Caprice models) to revive the Camaro. The LX platform (loosely based on the LH) followed up a few years after the American Ford equivalent platform with the Challenger. (The Australian Ford equivalent platforms, notably the EA169 and E8, never marketed any cars for North America, most likely due to Premier Automotive Group and cost-effectiveness; basically the same reasons why Ford never slotted a true sports car between the boring Thunderbird and the exotic GT to rival the Corvette and Viper).
The Lincoln LS V8 Sport's direct rivals are the Acura 3.2TL Type S, Infiniti M45, and Lexus GS430; in addition though, it is far more enhanced than either of its Japanese rivals.
EDIT: The Jaguar S-Type R was the performance model, but I initially omitted mention of it because it wasn't from Lincoln AND as reported by Jaguar itself and some journalists, it wasn't trying to go after the German super-sedans like Audi RS6, BMW M5, and Mercedes-Benz E55 AMG; it was really a slightly more enhanced Jaguar S-Type 4.2 slapped with a supercharger and a still rather soft interior; the R was faster than the 4.2, but only barely (and equally as fast, quick, and subtle as today's Acura TLX Type S, Kia K5 GT, and Toyota Camry TRD V6); the 4.2 was just a stroked 4.0.