Fact #1: Beethoven was a motivic composer. This means he would take a small and seemingly ordinary little note pattern (a motive or motif) and create a whole movement (or even a whole symphony) out of that one little idea. By repeating it, expanding it, condensing it, layering it, breaking it, transforming it, turning it upside down, inside out, backwards, etc, Beethoven would create something that was unified and diverse at the same time. One commentator rightly compared it to a mosaic - a beautiful image that is created from similar little chunks of tile. For example, his Fifth Symphony is built on that famous "Da-da-da-dum" motive at the beginning. Such creative genius won him instant recognition which leads us to our next point.
Fact #2: Beethoven was instantly popular, at least as a musician. He was notoriously rude, ill-mannered, and egotistical but the musical elite of Vienna still loved him. This translated to sponsorship. Many struggling musicians must compose pieces for other purposes such as dance music, church music, or requiems for the dead. (For example, Mozart never had a permanent commission). But Beethoven was free to compose as he wished. This leads to our next point.
Fact #3: Beethoven's music was self-expression. Essentially free from financial constraints, Beethoven created a music that was, above all, a unique expression of what was going on in his heart. He had a difficult childhood which included an abusive father. In fact, Beethoven fantasized for years that he was, instead, the illegitimate offspring of a Prussian king. As noted, he had very few social graces and his homes were veritable pig-styes. He was not handsome and did not care about his grooming, hence the now iconic wild hair. But the greatest challenge was his growing deafness. This considerable handicap haunted him and caused him to avoid people even more. Disturbed and isolated, music became Beethoven's therapy, if you will. Indeed, in his music I think you can hear the anger, the sadness, the struggle, the courage, the hope, the triumph, and more. A lesser man might have caved-in, but Beethoven overcame his many challenges, channelled that angst, and forged a body of work like no other.
There is much more to this celebrated composer but these simple observations will begin to help you understand why his music was so unique, innovative, and inspirational. I sense that Mozart is like a fine wine: smooth, refined, appropriate for all occasions. But Beethoven is more like hard liquor: fascinating, often harsh, with a much bigger kick. It's true he stands, historically, between the High Classical style of Mozart and Haydn, and the Romanticism of Mendelssohn, Chopin, and Brahms, but, truly, . . . he stands alone.