Let $W_{t}$ be a Wiener Process (a Brownian Motion starting at $W_{0} = 0$). What is the difference between $W_{t}$ and $\sqrt{W_{t}^{2}}$?
Using the Ito formula (in differential notation), $dW_{t}^{2} = dt + 2W_{t}dW_{t}$, so $d\sqrt{W_{t}^{2}} = [0 + (\frac{1}{2})(-\frac{1}{4}(W_{t}^{2})^{-\frac{3}{2}})(2W_{t})^{2})]dt + [\frac{1}{2}(W_{t}^{2})^{-\frac{1}{2}}]dW_{t}^{2} = (0)dt + (\frac{W_{t}}{\sqrt{W_{t}^{2}}})dW_{t}$, which doesn't really help (we need to assume they are different to show they are different, and visa versa)...
Edit: Of course they cannot be the same process, since $\sqrt{W_{t}^{2}}$ can never go below 0. But then why does it look like $E[\sqrt{W_{t}^{2}}] = E[\int_{0}^{t} \frac{W_{t}}{\sqrt{W_{t}^{2}}}dW_{t}] = 0$?