If you're making a pile of books or boxes, it's best to put the heavy ones on the bottom. The pile is "more stable" that way, in less jeopardy of overturning. We can apply this principle to the atmosphere to help explain why spring's weather tends to be more volatile than autumn's. Just think of the "heavy stuff" as colder air, while warmer air tends to be lighter.

In spring, the lower atmosphere warms quickly from the chill of winter, while higher up, the air remains relatively cold. This "warm under cold" layering - that is, light-under-heavy - isn't very stable. You get lots of overturning - rising warmer air and sinking colder air. And rising air lead to showers and thunderstorms. Plus, when air moves up in one place, and down in another, it also has to move sideways in between to balance it all out. And that sideways-moving air is wind. So spring tends to be pretty windy as well.

In contrast, in autumn, the lower atmosphere cools off quickly, but the warmth of summer lingers longer in the air higher up. The air near the ground is still warmer than the air at high altitudes, but the difference isn't as great as in spring. So there's less rising and sinking air going on. That generally means fewer showers and thunderstorms, and less wind. It's just a "more stable" situation.