What are the real and practical differences between the student t distribution and the Cauchy distribution? The results I get when I use them as priors in Bayesian linear models appear to be interchangeable.
Student's t-distribution becomes the Cauchy distribution when the degrees of freedom is equal to one and converges to the normal distribution as the degrees of freedom go to infinity. The primary distinction is that for either one or two degrees of freedom, then there is no defined variance for Student's distribution. This allows you to model the existence of a variance or not by choosing the number of degrees of freedom. With the Cauchy distribution, the scale parameter is not identical to the variance nor does it have a defined variance.
For small degrees of freedom and a very large sample size, the difference between a Student's t prior distribution and a Cauchy prior distribution may not be different enough for computational differences to arise.