Straw cowboy hats are typically worn in the hot summer months as they are lighter and allow a breeze, while felt hats are traditionally worn in cooler winter or fall months. The felt hat is more durable, but it does not allow air to flow through it, which makes it retain heat from the sun longer than straw cowboy hats.
A cowboy hat is a style of wide-brimmed hat that became popularly associated with "cowboys" of North America. The hats come in a variety of styles but almost always have a high crown and are made out of straw or felt. The wide brim of the hat, similar to that of the Mexican sombrero, is meant to create shade and protect the wearer from the sun because cowboys traditionally worked outside most of the day. The hats were originally made out of straw and worn when working outside, but the commercialized "Boss of the Plains" Stetson cowboy hat was made out of felt, which led to the rise in popularity of more formal felt hats.
Felt hats, like the original Stetson, are waterproof and provide good insulation. For this reason, they are often worn in the rain or in colder months. Because cowboy hats have become a cultural symbol, felt hats are also more popular as a formal hat when out in public.