The functional comparison
A standard quartz watch is accurate to about ±15 seconds per month; high-grade thermo-compensated quartz (Grand Seiko 9F, Citizen Chronomaster) goes to ±10 seconds per year. A COSC-certified mechanical watch is rated -4/+6 seconds per day; in-house chronometer movements (Rolex Superlative, Omega Master Chronometer) typically run at -2/+2 seconds per day. By any measurable standard, even the cheapest quartz outperforms the most expensive mechanical watch on accuracy by a factor of 30-100x.
What mechanical does better
Mechanical watches do four things quartz cannot. They hold value: a Rolex Submariner from 2010 is worth more today than a 2010 quartz Tag Heuer. They are serviceable indefinitely: a 1960s Speedmaster can be brought back to spec by any competent watchmaker, while a 1990s quartz watch with a dead movement is often unrepairable because the original module is no longer made. They have character: the rotor sweep, the second hand's tick rate, the winding ritual; the watch is alive in a way a quartz isn't. And they signal craft: the buyer is paying for handwork, not function.
What quartz does better
Quartz wins on accuracy, price, robustness, and convenience. A Casio G-Shock at $50 will survive 100 metres of water and a fall down concrete stairs; the equivalent mechanical dive watch costs $2,000+. A solar-powered Citizen Eco-Drive doesn't need batteries or winding for the life of the watch. Smart watches add fitness tracking, notifications, and contactless payment. For pure utility (a watch to wear in the gym, on a sailboat, while hiking), quartz is the obvious answer.
The high-grade quartz exception
Grand Seiko 9F quartz is one of the most over-engineered timekeeping mechanisms ever made: a brass pendulum-style impulse, hand-polished hands, and ±10-seconds-per-year accuracy. The Citizen Chronomaster Series 0 holds ±5 seconds per year. These watches outperform every mechanical watch ever made on accuracy, but the market still values them below mechanical Grand Seikos because the industry has spent decades positioning mechanical as the luxury-tier marker. There is no rational basis for this pricing; it is purely cultural.
What you should buy
If you have one watch slot in your life and need it to work for everything, buy a quartz: a Casio G-Shock for tool use, a Grand Seiko Heritage Quartz for dress, an Apple Watch for connected use. If you want a watch as a hobby or a daily-worn object that you enjoy, buy mechanical: a Tudor Black Bay or Seiko mechanical sit at the entry tier; Rolex Submariner or Omega Seamaster at the luxury entry tier. The two purchases are answering different questions.