Both types store energy in a mainspring and release it through the same gear train, escapement and balance. The only difference is how the mainspring is wound.
Manual (hand-wound) watches are wound by turning the crown, usually once a day. Without a winding rotor, the movement is often thinner, and the back of the movement is fully visible through a display back. Many owners enjoy winding as a daily routine.
Automatic (self-winding) watches have a weighted rotor that swings as the wrist moves. It winds the mainspring through reverser wheels or pawls. A slipping bridle stops the mainspring from being overwound. If the watch is not worn, it stops once the power reserve runs out. Most automatics can also be wound by hand.
Which is more accurate? Neither is accurate simply because of how it is wound. Accuracy depends on the quality of the escapement, the adjustment and the condition of the watch. Automatics can keep the spring more evenly wound when worn every day.
Quartz is different. A quartz watch uses a battery and a vibrating quartz crystal and has no mainspring. It is not a mechanical watch.