MLE for choosing models let say I have 2 models with the same number of parameters.
I implemented the Gaussian Log-Likelihood in R, based on real data for both models.
In the end, one model provides a higher MLE than the other one.
I conclude that the model with the highest MLE better fits with data and so is more relaible.
Is this intuition correct?
Thanks

Since your two models has the same number of parameters, comparing them by max likelihood will give the same result as comparing them with AIC (or other information criteria such as BIC). We can assume the two models are  not nested, otherwise they would be equal (or reparametrizations).
So if it is problematic to compare them by likelihood, the same problems would occur comparing by AIC. And there are indeed some problems, and also controversies. There are many posts here discussing this, so have a look there!
But, in many cases should be OK. What to look out for:

*

*The models should be for the same data, represented in the same way. So one model for $Y$, another for $\log Y$ is not comparable this way.


*If the likelihood functions are not of the same form, they must not be simplified by leaving out "irrelevant" constants! (R, for instance, often leave out constants ...)


*If one model is discrete, and the other a continuous approximation, likelihoods can not be compared.
