While Baroque and Classical music historically refer to two distinct periods of time (the former from around 1600 to 1750; the latter from 1750 to about 1820), and while they represent different styles, Baroque is often considered a subcategory of Classical. Be that as it may, they are distinct, and represent noticeably different characteristics.
Baroque music, which followed the Rennaissance period and represented a break from that earlier time, and which included Bach, Handel, Vivaldi, Telemann, Pachelbel, and others, involved an increased emphasis on the use of the music to support singing, in which respect it laid the foundation for opera. Instruments that did not lend themselves to a supporting role for the singers were relegated to a lower status within the orchestra, and those, notably strings, were elevated to a more prominent place. In addition to the emphasizing the role of strings, Baroque music also relied heavily on brass, specifically, the trumpet. [A fine contemporary production along this line is Wynton Marsalis' "Baroque Music for Trumpets," 1988] Trumpet, as the name is often applied, heralded, or trumpeted, the arrival of a movement or passage.
The Classical period continued the emphasis on strings, but rejected the Baroque period's tonality. As the Baroque era represented a rejection of the Rennaissance period, so the Classical era represented a move away from the Baroque. Mozart, Beethoven, and others of this period found Baroque music as being too dramatic and self-expressive. Mozart's struggles for acceptance with the establishment of his time are legendary, due in no small part to the popularity of the play and film "Amadeus," but those struggles did occur, although Salieri was treated a little harshly in the fictionalized accounts.
Classical music ushered in the era of sonatas, which set the tone for the various movements within a composition. With the rise of the piano's place within the orchestra, Classical compositions moved the emphasis to that instrument, while also adding more woodwinds than were commonly used in Baroque music.
Whether Classical music sounds "smart or sophisticated," and whether it is easy to follow, is entirely a product of the listener's skills in "hearing" key passages and uses of individual instruments and sections, and also, of course, in the individual tastes of the listener. Classical music enjoys a broad following around the world precisely because millions of people enjoy it. And, there is indisputably an intelligence or sophistication to much classical music. For this listener, Beethoven and Mozart were brilliant composers, although the former is, in some ways, "easier" than the latter.