The list of assumptions of the Gauss–Markov theorem is quite precisely defined, but the assumptions made in linear regression can vary considerably with the context, including the data set and its provenance and what you're trying to do with it. In theoretical accounts one often sees assumptions that errors are normal, independent, and homoscedastic, but that is far from always the case. Those assumptions justify the conclusions that estimators of coefficients have Student's t-distributions or that certain test statistics have F-distributions.
The Gauss–Markov assumptions are weak:
- the expectation of each error is $0;$
- the errors are homoscedastic (this is far weaker than identical distribution, let alone normality);
- the errors are uncorrelated (this is weaker than independence).