That is absolutely incorrect. What I hope you meant to say is that "dissonance and atonality are different". For instance, Walter Zimmermann's Wustenwanderung is atonal but primarily consonant, while Liszt's Etude "Ab Irata" is tonal but dissonant. Both tonal and atonal music can be primarily consonant or primarily dissonant. Of course Schoenberg's music is atonal, and of course it is dissonant; it uses unresolved, unstable chords.
Well it is really a question of semantics and context. The expression, "Schoenberg emancipated dissonance", refers to his conception of dissonance and consonance. Here's what he wrote in his theory textbook:
"...the more immediate overtone contribute
more [to the body of the tone], the more remote contribute
less. Hence, the distinction between them is only a matter of degree, not of kind. They are no more opposites than two and ten are opposites, as the frequency numbers indeed show; and the expressions 'consonance' and 'dissonance,' which signify an antithesis, are false. it all simply depends on the growing ability of the analyzing ear to familiarize itself with the remote overtones, thereby expanding the conception of what is euphonious, suitable for art, so that it embraces the whole natural phenomenon."
As far as unresolved chords, chords can only be considered unresolved if there is something to resolve into, in other words if they operate in a functional system. I can't think of a single phrase in Schoenberg (and I've played almost everything written for piano) that would sound good if it ended in a Classical triad. The idea is just absurd when applied to his music.
Are they unstable chords? That requires more subjectivity. I think he has unstable textures, but his harmony is like Bach's in that it all comes from intertwining voices (polyphony) rather than coming in blocks, like in Liszt or Handel. Depending on the action of the combination of voices - whether they agree with each other, or contradict each other, or create confusion - stability and instability is sensed and defined.
Lots of people have tried to make analogies between his efforts in twelve-tone composition with functional harmony, but they never hold up completely. There just isn't a polarity between dominant and tonic, the essence of functional harmony, in his music, no matter how you manipulate the tonal rows. Other factors are at play!
Walter Ramsey