Maximum likelihood estimation vs calculating distribution parameters "manually"
I think you are asking about how estimates for these parameters from your data sample (obtained by calculating moments) would be different from maximum likelihood estimation. If so, check out:
https://stats.stackexchange.com/questions/252936/what-is-the-method-of-moments-and-how-is-it-different-from-mle
http://gradquant.ucr.edu/wp-content/uploads/2013/11/Maximum-Likelihood-and-Method-of-Moments-Estimation.pdf