In brief, comparing two nested models with residuals which obey the same distribution, for example Gaussian, using a likelihood ratio test with a specific significance level can be considered as comparing two models with the AIC.
AIC scores are compared by differences so that the model is chosen which minimizes the AIC and thus, the model zero is accepted if $AIC_0-AIC_1 < 0$. This can be rewritten using the formula for AIC in the form $ 2L_1 -2L_0 <2k_1-2k_0$, which is $\chi^2$ distributed with $k_1-k_0$ degrees of freedom. That is actually a likelihood ratio test, substracting the log-likelihoods. The significance level can be determined by the critical value $2k_1-2k_0$ and a $\chi^2$ table.
This interpretation of the AIC is only possible when the models are nested. A nice overview about the similarities and differences between information criteria and hypothesis testing can be found for example in
Leontaritis, I. J., and S. A. Billings. "Model selection and
validation methods for non-linear systems." International Journal of
Control 45.1 (1987): 311-341.