The term "better" may be lead this discussion in a variety of directions, I think. Here are my thoughts, FWIW:
Quartz can be descibed as such in that it can be used to produce a watch that is more accurate, lighter, can be less expensive and contain many more complications than a mechanical automatic model. These aspects would allow for a very wide range of watch styles and versions, making possible the manufacter of activity-specific functions for every need.
But those quailities could also lead to the possibility of malfunction in those complications and the pesky need for periodic battery change requiring more frequent case openings. I wonder, too, if there isn't differences in accuracy among the various manufacturers and price points?
On the other hand, (wrist?), the beauty of the construction of an automatic may lend an "artistic" value. The feel of wobble or slight vibration provided by the rotor in some engines and, where possible, the visual appreciation of the fine workings of the machinery are sources of amazement which provide added pride of ownership, IMO.
These dimensions could be considered as making this watch type as better for their emotional impact. However, there is the additional expense of cleaning, proper maintenance and initial cost.
That being said, I believe we need both technologies.