Manuals do worse in all fuel economy tests against a modern 6+ speed automatic, especially if the automatic is a DCT system. They have been proven to do worse in performance driving as well. This has been proven by many tests by many entities over the years.
Engaging sport mode basically encourages the car's powertrain to favor acceleration and power over fuel economy and smoothness. For instance, if you are accelerating from a stop in sport mode, the car will hold each gear a bit longer than it would in regular mode. This helps the engine reach its maximum power output, which, for your car, falls roughly between 4000 and 6000 RPMs.
Fuel economy will likely suffer somewhat when using sport mode because the engine will spend more time at higher RPMs than it would in regular mode at the same speeds.