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".
Carmel Symphony Orchestra was created in 1976.
Dallas Symphony Orchestra was created in 1900.
Stavanger Symphony Orchestra was created in 1938.
Bern Symphony Orchestra was created in 1877.
Knoxville Symphony Orchestra was created in 1935.
BBC Symphony Orchestra was created in 1930.
Szeged Symphony Orchestra was created in 1969.
Sault Symphony Orchestra was created in 1956.
New Symphony Orchestra was created in 1991.