Assume that we have the following process:
$$X_{t}=\int_0^t \mu \, ds+\int_0^t\sigma_s \, dW_s $$
And assume that we observe $$Z_i^n=\sqrt{n}\left(X_{\frac{i}{n}}-X_{\frac{i-1}{n}}\right) \sim N \left(\frac{\mu}{\sqrt{n}},\sigma^2\right)$$
If we estimate the volatility from ML:
$$ \hat{\sigma}^2=\sum_{j=1}^n\left(X_{\frac{i}{n}}-X_{\frac{i-1}{n}}-\frac{\mu}{n}\right)^2 $$
And the realized volatility is given by $$RV_n=\sum_{j=1}^n \left(X_{\frac{i}{n}}-X_{\frac{i-1}{n}}\right)^2$$
My question is why they differ from each other?