Running shoes are designed for forward motion and to absorb the impact forces that running exerts on the foot. Training shoes usually have non-marking soles for indoor use and are designed to impart stability even during side-to-side stepping motions.
The soles of running shoes are flexible, lugged and often made of a stickier rubber to help grip the pavement or trail. Training shoes have comparatively harder and stiffer soles that are usually non-marking for indoor surfaces and are not lugged to enable a wider range of movement.
Running shoe uppers are flexible, lightweight and breathable; all things needed to allow the runner to participate in her sport more comfortably. Training shoe uppers are usually more robust, often made of leather for durability, protection and style.
Running midsoles often feature added support devices to correct for misalignment in the foot and are made of very soft, cushy substances. Training shoes are made of stiffer, more stable rubbers and are designed to support the foot in side-to-side motions as well as forward movement.
It's best to get a pair of running shoes for running and training shoes for doing weights and working out in the gym. A training shoe is too hard, heavy and stiff to work as a proper running shoe.