Maximum Likelihood
Is a generalized statistical framework for optimizing models
- Which means it is somewhat vague.
The best set of parameters is the one for which the probability is the greatest that the experiment performed would result in the measured values.
The character of the errors in your model are specifically built into the target function.
If the errors in the experiment and the model obey certain assumptions Least Squares is the proper Maximum Likelihood method for the problem.
Generally, the larger the errors the less applicable Least Squares becomes.