Mechanical watches have several advantages over quartz, battery-powered watches. These include: Product lifespan, craftsmanship, value retention, and socioeconomic brand perception.
Product lifespan - While a quartz watch might last 10-15 years with battery changes, assuming the purchase of a high quality timepiece, a mechanical watch made over a century ago can be found in perfect operating condition today. The reasons for this are simple. Mechanical timepieces can be serviced and repaired while many quartz watches cannot for the simple reason that while their components are mass manufactured, they do change over time and are phased out of production. The parts for many mechanical timepieces have also been phased out of production, but can in most circumstances be recreated. Not so for quartz watches. Note that most quartz watches are also made with the cheapest possible components, which are more prone to failure and breakage. A mechanical timepiece is designed to last a lifetime or two or three and can have a price to match. 99.99% of quartz watches are meant to be disposable consumer goods to be cast off at the first sign of problems or wear.
Craftsmanship - As the 1970's saw the Quartz crisis enfold the world and the near collapse of the Swiss watch-making industry, the Swiss responded by consolidating production and moving firmly upmarket. A quartz watch might be fine as a tool or a toy, but the upper class should enjoy the finer things in life, etc... This emphasis on mechanical timepieces as luxury goods led to dramatic per unit cost increases and a laser focus on high-end materials and flawless manufacturing and craftsmanship at any cost.
Value retention - While the vast majority of wristwatches never appreciate in value, including mechanical watches, there are a very few brands, such as Rolex and Patek Philippe, that are seen as a viable investment opportunity. At the end of the day, a vintage Patek might fetch anywhere from $10,000 to $10,000,000 depending on model, condition, and rarity.
Socioeconomic Brand Perception - One of the driving factors in mechanical watch sales, the perception of mechanical watches as a luxury status symbol, hello Rolex, Omega, Tag Heuer, Patek Philippe... If you want to make a good impression in many professional and social circles, a status watch like a status automobile can project power, wealth, and success.
Lastly, and not important enough to warrant major attention, Mechanical watches are immune to electrical interference, power loss, EMPs, etc.
As a counterpoint, the advantages of quartz are: Cost and Accuracy.
Cost - A quartz watch can cost $5. Enough said. A high quality quartz watch made to similar standards as luxury mechanical watches can be had for $1,000-5,000.
Accuracy - Most quartz watches are accurate to within 1 second per month. High-end thermo-compensated quartz can be accurate to within 1 second per year. Then there are the radio connected quartz watches that keep perfect time in accordance with atomic time signals. A mechanical watch is said to have "chronometer" worthy accuracy if it stays within two seconds per day and the majority will have variance of 10-30 seconds per day.