When flying in an airplane, or driving over a mountain pass, you've likely noticed your ears pop as you change altitude.
That's due to changes in air pressure. Air exerts more pressure at lower altitudes than higher ones. Even though you don't see it, air has a weight to it.
Think about the column of air directly overhead of you. As you go higher and higher in altitude, there's less air above you, so the cumulative "weight" of air over your body decreases, translating into a drop in air pressure.
You'll notice the quick changes in pressures because, in the case where you are gaining altitude, the air pressure inside your body will then be higher than the now-lower pressure outside your body.
When your ears "pop", that's the sensation of the body equializing the pressure difference between the air outside your body, and within.
It also works vice versa-- your ears will also pop when you experience an increase in pressure. This is why flying when you have a stuffy head is not pleasant because it is more difficult for your body to equalize the pressure.
Incidentally, when you do fly, you are flying at an altitude of between 25,000-40,000 feet. But inside the cabin, planes are pressurized to a relative altitude of around 7,000-8,000 feet -- or roughly the altitude of Santa Fe, New Mexico. (So, do Sante Fe residents not even notice the change?)
However, Boeing's 787 and its new fuselage design can allow for pressurization down to about 6,000 feet, supposedly making the flight more comfortable for passengers. (If any Boeing engineers want to weigh in on this in the comment section, that'd be great.)
So, why don't your ears pop during a storm?
Atmospheric pressure does change on stormy days, especially in the hours just before and right after a good-sized storm blows through. But even then, the difference in pressure is relatively small compared to the rapid ascension of flying or driving over a mountain pass.
A good storm might have a pressure range from 29.20 to 29.90 inches of mercury, but that roughly equates to a change in altitude of about 600 feet, and it's spread out over a few hours, so the change is pressure is much more gradual and not really that noticeable.