Training shoes are for exactly that...Training.
Some people have a habit of wearing running shoes in the gym and that is to be avoided if possible. Assess the type of exercise and activity you do each week and if much of that is in the gym then you need to select training shoes.
Training shoes offer greater stability when you lifting or squatting and will enable the power delivery you need to improve your PB's.
Another plus for training shoes is the wrap around comfort. If you need to strap onto a bike or row machine or do burpees or side planks you need the training shoe to be padded also around so you can focus on the exercise.