Beethoven did write another finale for it in 1798 for performance in Prague, but that is not the finale that it was published with. It was used by the composer as a vehicle for his own performances as a young virtuoso, initially intended with the Bonn Hofkapelle. It was published in 1801, by which time he had also published the Piano Concerto No. 1 in C major, although it had been composed after this work, in 1796 and 1797.
The B-flat major Piano Concerto became an important display piece for the young Beethoven as he sought to establish himself after moving from Bonn to Vienna. He was the soloist at its premiere on 29 March 1795, at Vienna's Burgtheater in a concert marking his public debut. (Prior to that, he had performed only in the private salons of the Viennese nobility.) While the work as a whole is very much in the concerto style of Mozart, there is a sense of drama and contrast that would be present in many of Beethoven's later works. Beethoven himself apparently did not rate this work particularly highly, remarking to the publisher Franz Anton Hoffmeister that, along with the Piano Concerto No. 1, it was "not one of my best." The version that he premiered in 1795 is the version that is performed and recorded today.