Do the adjectives "heavy" and "hard" mean the same thing when applied to music? What about their putative opposites, "light" and "soft"? In my view, they do not.
Heaviness is what distinguishes metal from other types of music. Hardness is what distinguishes rock'n'roll from other types of music. I've long argued that metal is both larger and smaller than rock. Some metal is rock, and some rock is metal, but much metal is not rock, and much rock is not metal. The same goes for "heavy" and "hard."
One easy way to define heaviness is simply the presence of a persistent and aggressive bass sound. But rap and some non-heavy rock (like Primus) make extensive use of bass, but we wouldn't want to call them "heavy."
I don't have good, precise definitions of "heavy" and "hard," but the following might be a start. Heaviness refers to whether the music has a dark, grim, or just serious edge. Hardness refers to the aggressiveness or "loudness" of the music. Music can be hard but not heavy. Yesterday I got in the mail Russell Allen's Atomic Soul, which is loose, playful, but very loud and aggressive hard rock. The guitars are in your face, but the riffs they play are loose and bluesy. Other really hard but not heavy bands might be Gorefest and Entombed (for the last few years). Today I got Royal Hunt's Eye Witness in the mail. The music is pretty soft as far as metal is concerned; crunchy guitars are not loud and dominant. But most of the songs are also not light, but fairly serious-sounding. I would consider Atomic Soul harder but lighter than Eye Witness. Another example of "soft/heavy" might be some parts of My Dying Bride's Turn Loose the Swans.
For these reasons among others, I think metal actually has more in common with other "serious" types of music than with glam rock and other light genres.
It's not as simple as minor versus major keys, either. Some indisputably metal stuff uses mostly major keys (Helloween, Freedom Call). But the juxtaposition of major and minor keys can have an "intensifying" effect (think of Evergrey's "The Great Deceiver").
Now, you can't take this definition of heaviness too far. What about really dark styles, such as darkwave, that we couldn't call "heavy"? Perhaps we need to modify the definition so that heavy music must be both dark or serious and somewhat "hard." In this construction, "heaviness" is a subset of "hardness." Still, the terms "heaviness" and "hardness" do not seem to mean the same thing.