Both a DSLR in live view mode and a micro 4/3 system camera use the main sensor for autofocus. However, on a DSLR it's embarrassingly slow.
This can't just be a software thing. How is it that one is so much faster?
Lens motors used for phase detect autofocus are optimized for a fast one-shot seek at a precise point and they go there with a good precision (with the chance of slight mistakes though); in layman terms they obey the command "go there, now!" in the fastest possible way. This because DSLR autofocus works by precalculating - using phase detect sensors - the distance and direction the lens focus mechanism should be moved.
Lens motors used for contrast detect autofocus, on the other hand are optimized for a succession of micro corrections of movements with very high precision and little inertia so to change direction quickly; in layman terms they obey the command "forward.. no no now a little backward.. no not this much now forward.. ok!" in the fastest possible way. This because focusing using the sensor works by moving the focus mechanism of the lens, see if contrast improves or worse and correct its action accordingly.
Note that there are also cameras using a mix of both technologies (e.g. Nikon 1), I don't know how the lens motors are built for those cameras.
Lens design (optimizing for continuous movement rather than fast jumps) is one factor but the main factor is simply that the main micro 4rds manufacturers (Olympus and Panasonic) spent a lot of time and effort on developing contrast detect autofocus technology (software and hardware) whereas DSLR manufactures tacked on contrast detect autofocus as an afterthought.
As proof of this, consider the Canon EOS M mirrorless camera. even with stepper motor lenses designed for CDAF, the M focuses much slower than M43rds bodies, simply because Canon lacks the CDAF expertise.