If you want the shortest possible answer: choose quartz if you value accuracy and want to forget about the watch between battery changes; choose automatic if you want a mechanical object that runs on the movement of your wrist and can be serviced indefinitely.
How each one works
A quartz movement runs on a battery. Current passes through a quartz crystal, which vibrates at a precise frequency, and that frequency is divided down to drive the hands. There is very little to wear out.
An automatic movement has no battery. A mainspring stores energy, a rotor winds that mainspring as your wrist moves, and an escapement releases the energy in regular increments. Every part is mechanical.
Accuracy
Quartz wins, and it is not close. A standard quartz watch typically keeps time to within a few seconds a month. A good mechanical movement is usually specified in seconds per day rather than per month. If your watch losing or gaining half a minute over a week would bother you, quartz is the honest choice.
Maintenance
Quartz needs a battery every few years and little else. An automatic needs periodic servicing, because the lubricants degrade and the parts wear.
The trade is that a mechanical movement can be repaired and re-lubricated for as long as parts exist. A quartz movement is often replaced rather than rebuilt.
Wearing habits matter
An automatic that sits in a drawer will stop. Most run for somewhere between one and three days off the wrist, depending on the movement and how fully it is wound. If you rotate between several watches, you will be resetting the date fairly often. Quartz does not care how often you wear it.
Which is more expensive
Mechanical movements cost more to make, so within one brand the automatic version of a model usually sits above the quartz version. That difference is about the movement, not about the case, the crystal or the finishing, which are often identical.
How to decide
Choose quartz if you want accuracy, minimal upkeep, a thinner case, or a watch you wear occasionally.
Choose automatic if you care about the mechanism itself, want something that can be serviced for decades, and will wear it often enough to keep it running.
Claude Bernard makes both. The automatics sit mainly in Proud Heritage, Classic, Aquarider and Slim Line; quartz runs across the range.
Common questions
Is an automatic the same as a self-winding watch?
Yes. The two terms describe the same thing: a mechanical watch wound by the motion of your wrist.
Do automatic watches need a winder?
No. A winder is a convenience, not a requirement. Wearing the watch, or winding it by hand, does the same job.
Which lasts longer?
A serviced mechanical watch can run for generations. A quartz watch will run reliably for years but is less often rebuilt when it eventually fails.
Is quartz less prestigious?
That is a matter of taste rather than quality. Quartz movements are more accurate; mechanical movements are more interesting to own. Both can be Swiss Made.