Atmospheric pressure is the weight that the atmosphere pushes down on you. There is actually a lot of weight on your shoulders, about 14 to 15 pounds per square inch on average for a person standing at sea level. The change in pressure, though, will affect the tissue around your joints, which may cause the pain.
So why do they notice it more during wet periods?
Often the atmospheric pressure changes at a faster pace closer to the center of low-pressure systems than around high-pressure centers.
Low-pressure systems typically bring clouds and precipitation, thanks to the convergence of air at the surface, which then forces air upward.
The winds, clouds and precipitation of the low are the reasons why we call larger low-pressure systems storms.
High-pressure systems mark the center of an air mass where characteristics of the air such as temperature and moisture are uniform. The air diverges from the center of the high, which promotes sinking air and nice weather. The arrival of a high normally brings a change in pressure, but because the weather is improving, there’s not as much of a reason for someone’s aches as there is with a low-pressure system.
A better association to link pain and weather would be wind. A change in pressure sparks off wind. In fact, the faster the change in pressure, the faster the wind will blow. If you want to predict when a pain may be coming through a weather forecast, keep an eye out for a day that is supposed to be windy.