Here's the differences between hard rock and metal as I see it:
Lyrics:
Hard rock lyrics are almost always about sex, drugs, and rock 'n' roll.
Metal lyrics are about other subjects: violence, fantasy, occult, politics....
Of course, both styles can steal subjects from the other side, but this division is true most of the time. I mean, when's the last time you heard an Aerosmith song that wasn't about sex? (Answer: "Amazing", 1993, about recovery from drug addiction.)
Music:
Hard rock songs are simpler, using only a few chords and using common chord progressions that have been around since the Beatles. (In fact, one of the greatest hard rock bands - Kiss - has stated that their main influence was the Beatles). Sometimes fancy guitar work will disguise the simple underlying chord progression, but the basic simplicity is almost always there.
Metal uses unusual scales and chords which lie outside the mainstream. The songs tend to be faster, longer, and more complicated. Metal generally requires more instrumental skill to play (though some hard rock guitarists, like Eddie Van Halen, are as skilled as any metal guitarist).
Attitude:
While some hard rock bands try to be genuinely innovative in their early years, most of them eventually find a style that they feel comfortable repeating on every album. AC/DC is the worst offender, though all hard rock bands suffer from this disease. Consequently, most hard rock bands are only interesting when they're young.
Metal bands place a higher priority on not repeating themselves, even at the risk of producing crap (see Metallica). This is why bands like Maiden can still amaze us after 25 years.
I should add that I wasn't considering any nu-metal bands when writing this post, because nu-metal is a lame joke (with only one possible exception, Korn's first few albums, but I still don't call that metal). I grew up in the era when long hair was a requirement to play metal. Unless your last name is Dickinson or Halford, I still think that's true. The nu-metal bands I've seen on MTV or VH1 look like spoiled frat boys who Motorhead would eat for breakfast.
(I guess that means I haven't been metal since 1997, when I gave in to adult responsibilities and cut my hair...)