None.  The instruments and repertory are equal.  The word "symphony"
means "reunion of sounds" in Greek (aside technical meaning in ancient
Greek music and various genres in occidental music).  The word
"philharmonia" means "friends of music".  In 18th, 19th centuries was
the societies for music recitals.  There was a little difference in
constituition: the Philharmonic Orchestra was a private organization and
the Symphony Ochestra was statal organization (they was a heritage from the
old orchestras of the courts).  Nowadays is indiffent the denomination.

Edson Ortolan
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