There are fundamentally very few differences between a hand-wind (mechanical) watch, and an automatic wind (auto-winder) watch. Both are ‘mechanical’ in the sense that they contain many moving parts and do not function by using a battery, though certain types of ‘kinetic’ movements take physical motion and convert it to electricity instead of directly converting it into kinetic energy via the mainspring. The use of their word ‘kinetic’ is somewhat misleading in this sense, but it is necessary to mention them as a sort of hybrid between traditional mechanical and automatic movements. Essentially if you remove the rotor on an automatic movement watch you end up with a mechanical (hand wind) watch.
The power output of a manual watch mainspring, as it unwinds, drops off linearly in the less expensive, unfinished movements. This changes the amplitude of the balance and thus the rate of the watch. This change is relatively small, on an ETA 2824-2 (Standard grade) it is only 20 sec over a 24 hour period (maximum). For a Top grade 2892A2 it is half that figure. This power drop off is not linear, but the first third is rather flat, then it starts to drop off.
An automatic watch, if constantly worn during the day and placed static during the night will be at, or nearly at, it full wind state during the day, and stay within a small range of the power curve. Then, at night, when at rest it will begin to wind down, but even after 8 hours would still be in the flat portion of the power curve. A manual wind watch will immediately begin to wind down and if wound in the morning, by night will have entered the second third of the power curve where it starts to drop off (most movements have a power reserve of 36 to 50 hours, which means the flat portion is 12 to 17 hours.
Aside from the "wear-and-forget" convenience and slight accuracy gains, there are some other reasons to prefer an automatic, such as if the watch is to be used as a diver's watch. With an automatic, the crown does not have to be manipulated as often, therefore, the crown seals will experience less wear and be less susceptible leaking. Also, since any motion in an automatic watch is converted to kinetic energy stored by the mainspring, you have a watch that is always ready and is not at risk of running down when you most need it.
With some more respect to accuracy, there are many variables that can affect the accuracy of a mechanical and automatic watch. If the hand-wind watch was only wound once every 24 hours then it might be expected to exhibit more variance than the auto-wound watch. However, if the wearer of the manual watch was "in touch" with his timepiece he could keep the mainspring within a narrow range and possibly beat out the auto-wind watch. If an automatic watch is allowed to run down to the end of its wind it begins to exhibit the same limitations and power fluctuations as its hand-wind cousins. Gravity can also play a role in that the direction that a watch is stored can cause micro-gravitational pulls on the oscillating wheel and cause a watch to speed up or slow down over the course of several hours. This technique is used often by mechanical watch aficionados to regulate their timepieces without having to open the movement and adjust it via the regulation lever.
The differences between mechanical and automatic time pieces are really more aesthetic than than are functional. Either type of watch will serve you for many years with a little bit of care.