Classical Music: Does everyone play the same notes in the orchestra while playing a symphony, or do composers write different notes for each instrument in the orchestra while creating their music?

Almost always different notes. Even Unison  passages mean that different instruments play the same note in different octaves.

Exceptions are so rare that this is the only one I can think of (though there are bound to be others). In Berg’s opera Wozzeck, right after Wozzeck kills Marie, Berg has an “Invention on a Single Note”. This has the note B played twice. The second time is in unison, with every instrument playing it in different octaves, in a long, terrifying crescendo.

The first time is usually played shorter, but I consider it even scarier. It is every instrument, one joining in after the other, playing the same note at the same pitch, B under middle C.