Robert Schumann called his Symphony No. 3 the Rhenish. In other words, it's about the Rhine, which for German Romantics was a kind of mythical river. Wagner set his whole Ring of the Nibelung around it. It was in the Rhine that the high-Romantic Robert Schumann chose to jump in his suicide attempt. So for Brahms the river and the memory of those years were joined. Likely that association of person and place is why the river and his memories around it came out in the symphony that begins with Robert's theme, and so with Robert himself. To some degree, this is a work in which, hidden in notes, Brahms looked back over his life.
Still, Brahms was not a composer of tone poems, and as far as one can tell most of his Third Symphony is largely laid out in abstract terms. The opening Schumann idea is more a tag than a theme; it only returns in the recapitulation and the coda of a first movement that mingles anguished sections3 with lyrical ones4. That mingling of pain and lyrical joy may be a telling matter, but it is not overtly tied to the triad of Robert, Clara, and Johannes.
The main theme of the second movement is hymnlike5. Clara wrote Brahms that it reminded her of a chapel in the forest. I suspect few people forget the first time they hear the third movement, with its yearningly beautiful, sui generis theme6.
The finale begins with an ominously murmuring gesture7 and returns to the anguished tone of the first movement. Even the contrasting second theme in C major has a detectably desperate quality8. Meanwhile Brahms has not forgotten the Schumann theme, which never quite found peace and resolution in the first movement. After a towering climax, the finale sinks into a long, quiet coda9 mainly based on a chorale theme that first showed up in the second movement. It is surrounded by flutters, and the coda adds up to a sense of autumnal reflection and recollection—of earlier parts of the symphony, and of its composer's life.
At the end of the symphony we arrive back at the Schumann theme, drifting downward in a magical, murmuring atmosphere10. At the end of the theme, over and over, we hear harmonies that suggest a phrase Brahms knew well, the "Farewell" motif11 that begins Beethoven's Lebewohl (Farewell) Sonata.
There is the autobiographical essence of the Third Symphony, written on the Rhine and recalling Schumann's Rhenish Symphony. Among other things, the Third is Brahms' farewell to all that, to the most exciting and agonizing period of his life, his Young Werther years, and his tragic mentor Robert Schumann.