I know I can only use the log-likelihoods of two models as selection criterion if they are nested. However, I don't understand this completely. Why isn't it possible to apply this reasoning to non-nested models? I think the higher the log-likelihood the better fits the model the data. I am also aware of other criteria as AIC but this doesn't helps me to understand the issue with the log-likelihoods.
You can compare the log-likelihoods of two non-nested models. See Cox (1961), "Tests of separate families of hypotheses"), Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., 1 & Cox (1962), "Further results on tests of separate families of hypotheses", JRSS B, 24, 2. But the log-likelihood ratio test statistic doesn't have the asymptotic chi-square distribution with degrees of freedom equal to the difference in the no. estimated parameters that is so convenient when the models compared are nested. See Wilks (1938), "The large-sample distribution of the likelihood ratio for testing composite hypotheses", Ann. Math. Statist., 9, 1.