Now you've opened a real can of worms
I'd really say that it comes down to a personal decision about your level of interest in watches.
Quartz watches are quite accurate and honestly are a set-and-forget deal. A quartz watch will need a new battery every few years, and ones designed for water resistance will also likely need some other minor maintenance at the same time. Low end quartz watches will generally die a slow death over 10-15 years. Better quartz movements tend to be in better cases and can be serviced and repaired just like a mechanical watch.
There are some quartz movements that, IMO, are works of art. The Rolex 5035 in the Oysterquartz(which has its own set of issues) is in that category, IMO. In general, though, a lot of quartz movements are boring.
It difficult to get a mechanical watch(whether hand wound or automatic) to keep time in absolute terms as well as a quartz watch. After a little bit of tweaking by a local watchmaker, the Rolex 3035 in my daily wear watch gains 10-15s a month, although it varies somewhat day to day in how much it loses or gains(depending-I've noticed-on how its worn. It does better on days where I'm more active).
There's also the whole aspect of them just needing more "fiddling." A hand-wound watch will need to be wound every day(honestly not a big deal, when I wear one I just get into the habit of winding it when I put it on in the morning). Unless you put it on a winder, an automatic will run down in 2-3 days(usually) of not being worn and you'll have to spend time setting the time as well as the day/date(if equipped) if you let it run down. I gave my dad an inexpensive(relatively) Seiko 5 Mechanical a few years ago, and he had issues keeping it wound because he would only wear the watch when leaving the house and often only left for a couple of hours at a time once or twice a day(these watches have no provision for manual winding, although the automatic mechanism is fairly efficient).
For someone who thrives on how watches work like me, there is no other option but a mechanic watch in some form. I'll also mention that there are collectors who have as much interest in quartz watches as I do mechanical, and would consider nothing but a high-end quartz watch.
For someone who just wants an accurate watch to wear and not really worry about, a quartz is a good choice. Many brands(including Tag and Seiko, off the top of my head) make similarly styled watches with both movement types.
There are also some modern "hybrids." Kinetic watches use an oscillating weight(like a mechanical) to keep a capacitor charged while wearing. These give all the accuracy of a quartz watch with little concern with running down a battery provided that the watch is worn semi-regularly. I don't know a whole lot about these, but think that the "winding" system is fairly efficient and the capacitors large enough that they will generally run for a couple of weeks without being worn.
More recently, you have a the Seiko Spring Drive which is basically a mechanical movement that uses a "braking wheel" combined with a quartz oscillator to regulate the timekeeping. An oscillating weight keeps the mainspring wound and supplies power to the quartz oscillator. These are as accurate as a quartz watch but also have a(truly) continuously moving second hand-a feature many people like in mechanical watches(although the second hand in mechanical movements isn't actually continuously moving-it just moves several times a second-usually 5-8).
To muddy things even further, there are also some "obsolete" electric watch technologies. There are balance wheel electric watches(Hamilton 500 series, probably most noteably) which are sort of a mechanical watch that uses a battery for power instead of a mainspring. These have many of the same "quirks" of mechanicals-including timekeeping ability-but without the reliability
. As I said, the Hamilton "electrics" are probably best known, although there are very few folks who even know how to service one(the "go to" guy-Rene Rondeau-just retired a few months ago).
The other notable ones are tuning-fork electric watches. Bulova Accutrons(old ones) are best known, although Omega and a few others made them. These uses a metal tuning fork that vibrates at a consistent frequency as their time base, and use a battery to power everything. These tend to be nearly as accurate as quartz, and also have a(perceptibly) continuously moving second hand.
