Do the commands logit and glm (Binomial family) in Stata use different fitting/estimation algorithm? In my mind, the commands logit and glm (Binomial family) in Stata both use maximum likelihood estimation, and the ML estimates and SE values are the same. But one instructor stated that they use different fitting/estimation algorithm. So I would like to hear the difference.
 A: Maximum likelihood is not an algorithm; it is an estimation method. Typically the algorithm used in implementing maximum likelihood should not matter, but it is always possible in difficult cases that some do better than others. 
In recent versions of Stata the default algorithm has been a variant of Newton-Raphson but users may choose other methods. 
See http://www.stata.com/help.cgi?maximize#algorithm_spec for an introduction. 
There should be no reason for anyone to bandy opinions on this. As of Stata 13, the entire .pdf documentation is on-line for all to see. 
A: A feature of GLMs that used to be useful was that one could estimate them by repeatedly using simple linear regression with smartly computed weights, so called iterated, reweighted least-squares (IRLS). So any statistical program that could handle linear regression with weights could estimate a GLM. This used to be a big deal when readily available programs for maximizing likelihood functions where less common than they are now. This is why some people still associate GLMs with IRLS. I guess that is the source of the confusion.
Notice, however, that IRLS and Newton-Raphson like algorithms are just different ways of finding the same maximum likelihood estimates. Also, as @NickCox remarked, the Stata command glm has the option to use IRLS to find the maximum likelihood estimates (by specifying the aptly named irls option), but by default uses a Newton-Raphson like algorithm.
