The S or Sport and Manual shift simply changes the gear aspects of shifting and allows the gears to either remain engaged longer and shift more crisply at different RPM ranges, or to allow you to manually shift at the range you choose within certain parameters allowable by the engine management system, such as a rev limiter. (Which usually only works in the acceleration mode, thus shifting down while at a high RPM can sometimes lead to disaster!)
The net output of the engine is the result of the gross horsepower at the crankshaft or flywheel, reduced by the loss through the drivetrain to the rear wheels. The only really good test of actual horsepower and performance I have found is by a chassis dynamometer that measures the output of the engine and driveline to the wheels. This is rather difficult with a 4WD system as most dynos are designed for 2WD systems only. The 375 SAE Certified HP in the 2013 LR4 is at the crankshaft or output shaft, and not rear wheel horsepower.
So, after that lengthy statement, your vehicle probably does feel and may be a bit quicker with it in Sport mode simply due to the faster shifts available under full acceleration mode. The actual diffference on, say a dragstrip, might be measured in hundredths of a second. But the real feeling that matters to most enthusiasts is measured on the seat of the pants. That extra kick when the gears change.
The final answer is that there is no extra horsepower available no matter which gearing set you choose. The engine wil output the same power; however torque peaks will change depending on the gear selected and when the shift occurs. That accounts for the kick in the pants you feel.
Cheers.
Pretty much spot on. However, the part I bolded may have been true 10+ years ago, but there are a plethora of AWD dyno's on the market today. Dynojet, Dyno Dynamics, Dynapack, Mustang, Maha... there are many and widely accessible. Engine tuning is non-existent in this particular market, but think of all the performance cars that are AWD - 911 Turbo, GTR, Subaru STI, Evo's, Lambo's, Audi's, BMW's, soon to be MB's AMG models etc.
Anyway, the shift from D to S just modifies throttle top-in, throttle mapping, and shift points. As magnum stated, what you have in terms of HP and Tq does not change in these beasts... it is what it is. If it was an older M model, then pushing a little button would have opened up a bit more power. Sadly, no little button for us.