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I have run negative binomial and quasi-Poisson models based on an hypothesis testing approach. My final models using both methods have different covariates and interactions. It seems that there are no patterns when I plot my residuals in both cases. Thus, I was wondering which test I could use to see which model fits my data better as the quasi-Poisson does not have any likelihood or AIC…

Also, I have a lot of overdispersion which makes me think that the negative binomial would be more appropriate, but I don't know if I can choose my model based on common sense…

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I see the quasi-poisson as a technical fix; it allows you to estimate as an additional parameter $\phi$, the dispersion parameter. In the Poisson $\phi = 1$ by definition. If your data are not as or more dispersed than that, the standard errors of the model coefficients are biased. By estimating $\hat{\phi}$ at the same time as estimating the other model coefficients, you can provide a correction to the model standard errors, and hence their test statistics and associated $p$-values. This is just a correction to the model assumptions.

The negative binomial is a more direct model for the overdispersion; that the data generating process is or can be approximated by a negative-binomial.

The quasi-Poisson also introduces a whole pile of practical issues such as it not having a true likelihood hence the whole stack of useful things for model selection, like likelihood ratio test, AIC, etc... (I know there is something called QAIC, but R's glm() for example won't give you it).

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