I think a fair way to explain the difference between the Baroque (Bach, Handel, Telemann) and the Classical period (Mozart, Haydn, the Beethoven’s early period) in one sentence would be: You cannot sing the Baroque. This is not entirely true, but accurate to a good degree. The Baroque was getting way too complicated and elitist and the guys from the Classical period wanted something simpler, something that was closer to people like you and me (not really, but you get the idea). That was called the Gallante music or style. This is evident when you compare Bach and Mozart. Bach had a bunch of melodies playing at the same time, “communicating” with each other (certainly not a small achievement), so good luck if they ask you to sing the melody(*). Mozart on the other hand has a clear and in-your-face melodic line; his melodies are easily recognized and remembered.
Since I have been listening to Mozart’s Piano Concerto in C major K.503 lately, I will post the first movement here. I am used to the Brendel version, but I found the Ginzburg version in YouTube. The modulations here are just out of this World, it goes as far as to modulate to C minor as soon as the first theme starts (the violins do that, it is around 1:32 on the video), only to go back right away to C major. Mozart could definitely modulate effortlessly, and that is what he does for the whole Concerto.
On a cool note, even when we tend to think about Classical music (in the broad sense, I am not referring to the Classical period) as something extremely rigid, the truth is that a lot of these guys loved to improvise. Mozart, who was famous for his improvisations, did not always bother to write down the cadenzas he played (as these improvised parts were called). That is the reason why you can find different versions of this Concerto according to whom you are listening to.
(*) I am in no way diminishing Bach by the way, I agree with the Rumanian philosopher Emile Cioran when he wrote:
Without Bach, theology would lack object, Creation would be fictitious, the peremptory nothingness. If anyone owes it all to Bach, it is, doubtless, God
(from “All Gall Is Divided”, I took the translation from here)