Typically the live view (contrast) based auto focus is slower. This is one of the reasons that many of us prefer a DSLR camera over smaller type of digital camera...The DSLR is capable of faster AF, which often relates to less 'shutter lag'. The biggest complaint about P&S cameras is that they take to long to snap the photo, and in many cases, that is related to their slow AF.
Case in point, some DSLR cameras that offer live view, give you the choice to use 'Quick Mode' AF, while in live view. In this mode, the camera flaps down the mirror, uses the phase detection AF, then flips the mirror back up. They offer this because it can be quicker/faster than using the contrast detection AF.
As for one or the other giving you a sharper image....I wouldn't put it like that. I might say that one mode or the other might struggle to reach focus in time...but if they are both working correctly, then they should both be able to achieve accurate focus, and thus the resulting sharpness of the photo should be the same.