What's the difference between harmony and polyphony? Is it a matter of perspective or is there a qualitative difference?
Harmony refers to the "vertical" relationship between simultaneous pitches in a musical texture (usually, but not always, chords - see below for the exception). However, it also refers to the "horizontal" relationships between successive vertical relationships of pitches; it's probably easiest to think of these as chord progressions. The exception, mentioned above, is that even a single melodic line will create a series of implied harmonies: sometimes this is very obvious, for instance a series of arpeggios; sometimes it is ambiguous, as the single line could suggest a number of harmonies. Also, a single line or two-part counterpoint can easily imply a series of chords containing three and four (or more!) note chords. (@Pat Muchmore mentions this in his comments.)
Harmony can be thought of as the framework of pitch relationships between any number of simultaneous elements in a musical texture, be they: melody, bass-line, block chords, multiple melodic lines, counter-melodies etc.
It is important not to confuse the concepts of harmony and accompaniment; harmony is not simply "padding" between the main melody (if there is one) and the bass part. However, the term harmony is often used informally to mean just this: parts of the musical texture that accompany a main melody part. (When writing arrangements I'll often mark parts as Melody, Harmony and Bass parts.) But, strictly speaking, all of these parts, accompanimental or not, define the overall harmony.
Polyphony is a related, but quite different concept. Polyphony refers to a texture containing a number of independent melodic lines (or voices, if you like). A polyphonic texture does not rely upon a series of "block-chords" (a homophonic texture) to reveal the harmonic progression of the music, but it still has a harmonic framework. Although independent to a certain degree (for instance, in terms of rhythm, contour, motif, intervallic character) these lines are still tied together by the overall harmonic framework. (A polyphonic texture still "has harmony".)
Although a polyphonic texture is to a large extent conceived and perceived as layers of simultaneous "horizontal" melodic lines, good contrapuntal writing must take account of the vertical relationships between parts, in other words, they must all fit within the ongoing harmonic framework of the piece.
Arguably, it is this very tension between horizontal melodic impetus and vertical harmonic relationships, that makes great contrapuntal (or polyphonic, if you like) music so satisfying to compose, perform and listen to.
As a closing point (and maybe one for others to comment or respond to): arguably the harmonic relationship between parts breaks down in certain types of polyphonic modern music, where intervallic/motivic relationships in individual melodic parts can override the relationships between parts. (Even so, it is usually the case that there is a significant amount of pitch organisation between simultaneous contrapuntal parts, even if they do not fit within a "traditional" harmonic framework.)
Harmony supports the melody. Polyphony is when there is more than one independent melody.
In the case of harmony, everything supports the melody. Their may be secondary melodies or the melody might switch between instruments, but there is not two independent melodies at the same time.