Please try to provide a photograph of the white fuzz.
Examples of common betta infections.
If it is...
White cottony growth on fin, mouth, tail, or the entire body essentially, and the fish remains still on the bottom of the tank, your fish probably has a fungus. You'll want to treat this immediately. The website I just provided will tell you to use Jungle brand medicine, but don't bother looking, they don't make them anymore. You'll probably be able to find API Fungus Cure
at the store you got your betta from. Go there ASAP and get the medication and begin treatment using the instructions from the package.
As the medicine in the package is designed for 10g aquariums, you'll have a few issues distributing the medication properly for a 1.5 gallon tank. Someone like Sakura would recommend you dissolve an entire packet of the medicine into 5 cups of water, and then add 3 quarters of a cup to the tank water to ensure the proper dosage, since trying to measure 1/16 of a teaspoon or less can be quite tricky. Perhaps someone else on this thread will be able to suggest another method of dosing.