The reason you want to have a different towel for the gym than a standard at home is because these towels are made differently. If you look closely, the ones that are offered by the gym you frequent are different than the ones you use at home. The reason for this is obvious, gym towels are designed for the tough needs that are experienced at the gym. These are not standard towels in that they have been constructed in a way to serve you better with double stitched hems especially.
One way that specific gym towels are different is that they absorb more water. They are generally 'fluffier' in nature and thus have a stronger holding capacity. As such, they can also be used for extensive periods and can be washed more times than the average towel without having their texture compromised.
A good towel will be made with cotton, due to cottons exceptional ability to absorb water. Cotton also lends to the durability of your gym towel, and can be carried easily in your gym tote. A mesh bag is a good idea to store your towel in, as this allows for air and ventilation to your towel, which will decrease the spread of germs and bacteria. While it is not a bad idea to use the supplies provided by your gym, having your own unique gym supplies will help you get optimum results in your unique gym experience.