Running shoes are designed for FORWARD motion and trainers are designed for quick, agile, LATERAL (side to side) moves while still being able to accommodate bursts of forward motion (ex.interval training). Running shoes have the level of cushioning need to absorb the shock of impact (foot hitting the ground/pavement) and while trainers do provide support, they're also built for flexibility and agility. If you compare the bottom of these shoes, the one on the right has a very distinct, sharp tread and the one one the left a very slight tread.
What's the danger of wearing the wrong shoe? Increased risk of injury which could leave you on the sidelines making it much more difficult to reach your health and fitness goals, so please listen up!