Water filter as opposed to bottled water.

I always had a water cooler and got the five gallon jugs delivered to my house. After I stopped that I would buy the gallon jugs and the 24 packs of bottled water. Just recently we got a water filter that clips onto the faucet and then one where you pour the water in top and it goes through a filter into a storage at the bottom with a spout, and the whole thing goes int he fridge. I think I can still taste a difference even though we are filtering the tap water twice...it might just be a mental block on my part because everyone else says it's fine. It's cheaper than buying all the bottled water...any input? Do you prefer bottled or filtered?