I second what Ludwig and whynot say above. As Brahms matured, his melody lines got longer and longer. It's not unusual in later Brahms to find 12-16 bar melodies. He also became increasingly chromatic, not to the extent of Wagner by any means, but frequently shifing harmonies that were often at least partly chromatically based.
He's capable of great power and immense tenderness. I think he is the most earnest of the composers, he and Schubert. He seldom departed from the classical forms, but the content was anything but classical -- very romantic, very rhapsodic, full of counterpoint and syncopation. He could be forceful but never harsh or angry like Beethoven can be. Brahms' chamber works are probably his best music and a great place to get to know him. It's too bad more people don't know this incredible music.
His late piano pieces, especially those he labeled Intermezzo, are all jewels of the small-piece repertoire. They are dense with complexity and meaning, often more than it seems they can bear, as though a bigger work was in chrysalis form. The somewhat overplayed Op. 118, No. 2 in A is one of the great small piano works of all time, a profound composition of yearning and longing with an undertow of sadness and regret.
I love his symphonies and concerti, too, all of them (especially the violin and double), but I think he really excelled in the intimate setting of the chamber work and piano solo.
Topic: for the brahms lovers!