Difference between asymptotic normalities of OLS and MLE Let's see the comparison below

Asymptotic normality is given by CLT for both cases. In MLE case, a variance of $\hat{\theta}$ is in distribution as $\frac{1}{I(\theta)}$, but in OLS case $\sigma^2Q_{xx}^{-1} n$ is not a variance of $\hat{\beta}$. 
The second case seems to follow CLT well, however both cases are already proved and easily seen anywhere.
What is a difference here? 
 A: It's not clear to me what you're asking here. Are you asking why the asymptotic distributions of the OLS and MLE estimators are different? If so, that's contingent upon distributional assumption.
The assumptions about the error term matter heavily in determining whether or not the OLS and MLE estimators coincide. When you make the assumption that the error terms are normally distributed, the $\hat{\beta}_{MLE}$ - the MLE estimates - are the same as the OLS, $\hat{\beta}_{OLS}$, estimates.
The MLE approach asks you to
$max_{\beta_{0},\beta_{1},\sigma^{2}} \; {L(\beta_{0}, \beta_{1},\sigma^{2})} = \prod_{i=1}^{n}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(Y_{i}-\beta_{0}-\beta_{1}X_{i})^{2}}{2\sigma^{2}}}
$.
This holds for small samples. When you have large samples, this holds generally, i.e. the MLE and OLS are the same in the asymptotic sense. Note that general result doesn't rely upon normality though. It just hinges upon the fact that the draws of the estimates approach normality with large enough $n$. So, the formulas you posted are the same but the MLE is written implicitly. 
