A major selling point of bottled water is that it is purportedly free of many of the contaminants that you might see in regular tap water. There is certainly nothing wrong with trying to drink water that is free of contaminants! You want to be in control of what goes into your body, and that is a healthy way of living. However, bottled water is not necessarily contaminant free, and in fact can be basically just the same as tap water!
A water filtering system is better at guaranteeing the removal of contaminants from what you drink. How do you know? Unlike tap water, this is expressly what water filtering systems are for. They list the contaminants that they remove and then remove them, whereas bottled water just offers water that is better than some tap, generally, without specifying much more than that. So, if you are looking to remove contaminants, and every tap is different, then getting a water filtering system is a good idea.
What really sets a water filtering system apart, though, is the ability not only to remove contaminants but to add minerals that are healthy. Europeans started drinking bottled water long before it became popular in the states, not because these bottles were pure water, but because they were filled with extra healthy minerals. Many water filtering systems allow you to control what minerals are added into what you drink, giving you even more control of what goes into your body.