Filters, like Brita, use an ion exchange to remove unwanted minerals (like lead, iron, etc) in exchange for safer ones (like calcium, potassium, etc). They may also include an actual filter to remove particles. But you'd have to determine what size of particulates are filtered out and what size are let through; and see if that filtration size will remove any harmful microbes that you may be concerned about. If your concerns are about the minerals that get dissolved in the water, then filtering is probably best. If on the other hand you are more concerned about getting sick from microbes, then boiling would be best. Ideally, distilling water (boil the water, then let the steam condense in an unreactive cooling coil) would remove not only the microbes, but also the minerals as well. However distilled water tastes very bland (the dissolved minerals actually give the water a little taste to it), so most people prefer ion exchange to drink rather than distilled water (and usually distilled water is only seen in research laboratories, where even trace amount of minerals / particulates can affect the results of experiments).
Since I get my water from the city, it's mostly the mineral content that I'd be worried about, so I usually drink filtered water. But even if you get well water, you may consider filtering in addition to boiling, esp. if you live in an area with a high mineral content (ie if you have "hard" water).