Not really... although the terms are generally used synonomously, these days.
An orchestra is defined as an ensemble of different families of instruments. So, that could include concert bands, big bands and the jazz orchestra (big band, 2 horns and strings), among others. A symphony is a specific musical form, usually played by a (fairly) specific ensemble. So, the term "symphony orchestra" is literally an ensemble capable of playing that kind of piece... thus the modern symphony orchestra. Generally, the term "symphony orchestra" is more specific, when talking about the classical ensemble.
Bournemouth Symphony Orchestra was created in 1893.
London Symphony Orchestra was created in 1904.
Nuremberg Symphony Orchestra was created in 1946.
Leeds Symphony Orchestra was created in 1890.
Baltimore Symphony Orchestra was created in 1916.
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The first symphony orchestra was in Germany.
They are interchangeable and have pretty much the same meaning. However, if one wants to get technical, a "symphony" is a composed piece that is played by the "orchestra".
Melbourne Symphony Orchestra was created in 1906.
Youngstown Symphony Orchestra was created in 1925.
Tulsa Symphony Orchestra was created in 2005.
Pittsburgh Symphony Orchestra ended in 1910.
Pittsburgh Symphony Orchestra was created in 1895.
Tokyo Symphony Orchestra was created in 1946.
Memphis Symphony Orchestra was created in 1960.
MIT Symphony Orchestra was created in 1884.
Portland Symphony Orchestra was created in 1923.