When contrasting classical music and impressionist music, let's assume "classical" refers to the music of Haydn and Mozart's era. With that being the case, these are two very different types of music.
The most obvious difference is tonality. Classical music draws on the tonality codified by J.S. Bach, that of major and minor scales. In contrast, impressionist music uses expanded tonality, where the chords are not so clearly major or minor. These composers stack notes upon notes such that there is much less of a sense of a key center.
A second difference would be instrumentation. Orchestras of Haydn's and Mozart's time had fewer instruments, and some of the instruments were not in their final form. For example, the French horn of the classical period was a natural horn (no valves). In contrast, impressionist music makes use of a wider range of instruments and the instruments are advanced enough that they are much less acoustically limited. Impressionist orchestrations reflect this fact.
Related to the differences in orchestras between the two periods, there is a contrast between composers' approaches to tonal color. During the classical era, composers did not "paint" with sound in quite the same way as the composers of the impressionist era did. You can hear this in the opening bars of Debussy's "Afternoon of a Faun," where the flute and the harp establish a rich sonority that signals the beginning of a story, the sonic equivalent of a play taking place in one of Monet's garden paintings.
Compare the musical elements of classical music and impressionistic music.
Comparing the musical elements of any two styles or eras of music is an intimidatingly broad endeavor.
The term classical refers to the symmetry, balance, and egalitarianism espoused by the artistic ideals of the Greeks, most notably their architecture. The music roughly coincided with the European Age of Enlightenment. Classical music may be defined as the musical language practiced by European musicians (FJ Haydn and WA Mozart being the most well-known) between approximately 1750 and 1825 (1824-1827 are also accepted dates). The dates are not arbitrarily chosen; 1750 marked the passing of J. S. Bach and Beethoven completed his Ninth Symphony in 1824. The previous era of music, known as baroque, lost its greatest practitioner with the death of Bach. This created a vacuum in which a new style of music began to flourish. This new style probably reached its zenith in the music of Beethoven (Beethoven's artistic ideas began to reach toward a new expression of music that would become known as Romantic). As with all forms of art, classical music was a reaction to a previous style; whereas baroque music was ornate, embellished, complex, and often coldly empirical, classical was sparse, streamlined, balanced, and often contained some emotional storm and stress (Sturm und Drang).
Impressionism, generally regarded as a style of visual art practiced most famously in France, does not have an obvious parallel in music. The French musicians Claude Debussy and Maurice Ravel were both professionally active during this period known as impressionism. Debussy's piano prelude entitled La Cathédrale engloutie (The Submerged or Sunken Cathedral) seems to exist within the same style as Claude Monet's Rouen Cathedral paintings. Pointillism (see the George Seurat painting A Sunday Afternoon on the Island of La Grande Jatte), one of the sub-genres of impressionism, can easily be heard in the music of Anton Webern. The pointillism heard in Webern's music would not have been possible without the contributions of a few key French artists who began to break away from the German/Romantic tradition in general and from Wagner in particular. Carmen, the opera composed by George Bizet, pointed the way toward a more realistic, natural style of music. The Paris Exposition of 1889 brought the sounds of the Javanese gamelan ensemble to the ears of many French musicians, including Claude Debussy. The ensemble's music, which disobeyed all of the European conventions of music, seems to have acted as a catalyst that that steered many European, especially French, musicians away from the accepted method of German Romanticism. This new style of music that coincided with the Impressionistic visual artists jettisoned overwrought emotional and dramatic content. The music attempted to create an atmosphere in which the listener could exist rather than an emotion for the listener to experience.