When you workout in the gym, you will be doing different types of moves and exercises. Contrary to running, which is a movement in the same direction; pretty much straight forward with the occasional turn around the corner. Your moves in the gym are more unidirectional, including stepping out left and right. And you will be putting quite some pressure in your shoes when you are lifting weights. Some rope climbing or other cross fit drills can also be part of the mix.
What is different about training shoes?
Because of the above exercises, training shoes need to provide more grip, support and protection than running shoes. Think of a more grippy outsole, low to the ground platform and reinforced upper materials.