According to the Gauß Markov theorem, the least squares estimator is the best linear unbiased estimator, given some assumptions.
The maximum likelihood estimator however, has asymptotically minimal variance, i.e. in the limit of large N it has the lowest variance amongst all unbiased estimators.
First I thought that this wasnt a contradiction as these two estimators are equal for a linear model with gaussian noise. However, the Gauß Markov theorem holds even if there is no Gaussian noise, i.e. in cases when the MLE is not equal to the LSE. Can someone help me to solve this contradiction?
There is no contradiction...even if you apply it asymptotically as you discuss in the comments under the question. The MLE is a general estimator that depends on the distribution and is not necessarily linear. The GM theorem applied to linear estimators.
The MLE is asymptotically the best among almost all estimators that are consistent and asymptotically normal (under minor regularity conditions). It achieves the asymptotic Cramer Rao lower bound.
The linear LSE is the best among all consistent and asymptotically normal linear estimators.