A Brita tap water filter isn't going to give you spot-free washing/drying. At the most, you will probably only have better tasting tap water now.:dance
Drinking water filters like the Brita are mostly for improving the taste of potable water thru a combination of carbon and mechanical filters. They have virtually no ability to soften hard water. They will remove the "chlorine taste", some organic chemicals, and small amounts of particulates or sediment.
A water softener cartridge/filter will reduce your water hardness which may help substantially, but only DI, RO or distilled water will give you anything close to spot-free water. Most water softeners use a resin based cartridge which exchanges sodium salts for Ca and Mg. The Ca/Mg in hard water form carbonates and sulfates that precipitate out of the water and they leave behind the tell-tale water spotting.