Brahms 3rd is a Beethoven symphony played backwards. It is a journey from light to darkness. The movement that Beethoven would put first is last, and the movement Beethoven would put last is first. Each movement is darker and more resigned than the one that came before until a last movement nightmare that is the one of the most violent things Brahms ever wrote. Instead of a Beethovenian peroration, Brahms ends the symphony ends the symphony with gentle resignation.
(World Champion Brahmsian Bruno Walter layin' it down with the pre-WWII Vienna Philharmonic. Sound ain't great, but neither of his remakes were on youtube. And this one might be even better than the other two. Nobody gets inside Brahms 3 like Walter.)
It's the hardest of the Brahms Symphonies to understand, and most conductors don't play it as though they do either. Some music is by and large musician-proof. Bach is a composer who generally works his magic so long as you play the right notes in the right order, so is Stravinsky. Brahms is nothing like that. His music needs a conductor willing to take a proactive hand, but not too proactive. The best Brahmsians know the music so well that they've internalized it. They unconsciously remember every harmonic ambiguity, every subtle metamorphosis of each motif, and know how to render all these ornate details into a coherent whole. Conductors like Toscanini and Szell were always a little too controlling for Brahms. While conductors like Furtwangler and Jochum were always a little too willful. The best Brahmsians are conductors like Felix Weingartner, Bruno Walter, Christoph von Dohnanyi and James Levine, who know exactly how much give and take to give each moment of the piece. Always precise, always flexible.