If I played the opening of this to you, would you have thought it was by Mozart? I'd guess probably not, I certainly didn't the first time I heard it (which actually wasn't too long ago). Maybe this is because of how we stereotype Mozart - perhaps we expect something akin to this piece's second movement, and yet we begin with something very different.

Perhaps part of the reason this piece seems so ahead of its time is that it is in a minor key, allowing composers of this style to incorporate more dissonance and therefore arguably more perceived tension into their 'traditional' use of harmony and counterpoint. Furthermore, this concerto begins in what some might call a 'Sturm und Drang' (storm and stress) style - although as Clive McClelland argues, this term is perhaps misleading as it wouldn't have been used at the time - as Sturm und Drang wasn't a thing. He prefers the term 'Tempesta', and I think this sums up the first movement rather well on a number of levels. If you're interested in this, I'll post a link below.

Another aspect which I think points this towards a somewhat Beethovenian musical schema is its use of sudden, and quite loud dynamics. It's far from gentle, as you might expect. Yet of course, there are calmer moments even within this movement - and when themes are transformed later on into a major key we see how Mozart has cleverly manipulated essentially the same musical material to mean two rather different things to the listener.

One thing I find quite useful to do when listening to Mozart, a composer who, I must admit, doesn't often immediately grab my attention, is trying to listen for his clever use of simple melody, and how the notes in the melody relate to everything else that's going on. This approach, I think, is quite useful for decoding the second movement. The genius here is that there is actually quite a lot of conveyed emotion, and the music actually does feel very sensitive. Chromatic decorations and simple (not at all appearing as frilly or elaborate) ornamentation add an emotional depth to the music that might not be initially perceived.

Finally, the third movement is designed to showcase the pianist's technical skill and end the piece in a flourish, bringing together the fiery nature of the first movement with the emotional depth of the second, as well as the intricate sense of controlled yet meaningful musicality that Mozart does so well.
 

McClelland's writing in the Oxford Handbook of Topic Theory: http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199841578.001.0001/oxfordhb-9780199841578