I have been attempting to estimate alpha and beta from a beta binomial distribution given my data. There are R packages like VGAM to do this. I am wondering if there is a difference between estimating these parameters for a beta binomial distribution and if I estimated them for a beta distribution. Is there a way to fit my data to a beta distribution and get the same alpha and beta as when I fit it to a beta binomial distribution?
Beta and Beta-binomial are two different distributions. Beta is a continuous distribution (values in the $[0, 1]$ range) while Beta-binomial is discrete (integers from $0$ to $n$). So it is not a good idea to use both distributions to approximate your data since it can be discrete or continuous, but not both.