Price and weight are the only advantages of using 1/4''. My last cooker was built from 3/8'' because it was free, it took a little longer to warm up all of the metal but once it was up to temp it was really stabile.
Nothing wrong with thicker but the difference between 0.25" and 0.3125" would probably not be real noticeable or worth the investment. Thicker metal will last longer and hold heat a bit better but comes at a higher cost and is harder to work with. In my opinion, 0.25 is kinda the " sweet spot" and works well for most designs. Nothing wrong with going thicker, tho....it's your money!