As mentioned in the discussion of PC No. 1, No. 2 was written ten years previously, between 1787 and 1789, when Beethoven was in his teens. He premiered the work himself in Vienna in 1795. While it had undergone many revisions, the version Beethoven played on that occasion is the version played today.
Freddy Kempf says:
I love no. 2—and have really warmed to it now after having play-directed it. I think it’s not often played because of how tricky it is to put together—it needs a very stable pianist and the 2nd movement is particularly difficult to get sounding really good. No. 2, I feel, is very much influenced by Haydn and the Baroque era. Beethoven idolised Mozart and came to Vienna hoping to study with Mozart but only ever met him fleetingly and ended up studying with Haydn. I feel that concerto no. 1 is Beethoven trying to grasp the new sound that Mozart had created, whereas No. 2 is the sound he’d studied and been handed down from Haydn. Both works still certainly give a hint of the romanticism to come—the slow movements really expressing very real, human sentiment.