A betta really needs a heated, filtered tank of at least 2.5 gallons to thrive.
Like any other fish, their waste produces ammonia, this ammonia is toxic to the fish, and needs to be removed. In a bowl, or another small, filterless tank, this ammonia has no where to go unless you remove it, which leads to frequent, if not daily water changes. A filter and nicely sized tank on the other had allows for the establishment of the nitrogen cycle. The nitrogen cycle is a cycle that removed ammonia from the water through the growth of beneficial bacteria. This bacteria grows in several stages. First, the ammonia is transformed into nitrite, then the nitrite needs to be turned into nitrate. Ammonia and nitrite in any amount can be deadly to the fish, but fish can stand a relatively high amount of nitrate. This process takes about a month to establish, but allows you to perform only weekly water changes of around 30%, and allows the fish to live much more healthily.