Barefoot running is gaining momentum, especially among athletes who think modern running shoes are detrimental to foot health. Truth to be told, high stack, carbon plate-enhanced running shoes indeed force runners to run in a particular way which might not be the most ideal for at least some percentage of runners out there.
That said, running barefoot takes some getting used to, especially if you usually wear cushioned shoes. Even big-name barefoot brands such as Vivobarefoot (retailer link) recommend easing yourself into barefoot running as it requires you to move very differently compared to when you run in cushioned shoes.
It's recommended to start running shorter distances – as in a couple of hundred meters – at first, then gradually increasing the distance as you get more comfortable. It's also beneficial to run on softer surfaces such as grass or sandy beach at first, as these might mitigate impact force better than landing hard on tarmac.
As for running in sneakers, these shoes lack support and cushioning, and by running in them, you can end up hurting your feet more than running barefoot.
In sneakers, just like in running shoes, your sole is far away from the ground. However, the thick foam doesn't provide anatomical support (as in the case of running shoes). If anything, it prevents you from being able to stabilise your body yourself, which will confuse and tire out muscles.