Seems that two of the most common materials for the soundboard of a classical guitar (also called tonewoods) are cedar and spruce. Regarding tone, is there a significant difference between the sound a classical guitar with a cedar top will produce, when compared with a spruce top? If so, how do these materials affect the sound differently? Which are their "tonal properties"?
It's not possible to generalize, since the overall tone is shaped by many different variables (as Tim mentioned in the comments), and in the specific case of spruce vs cedar, the difference seems to be subtle.
Given two identical guitars, one with spruce tonewood, and other with cedar tonewood, there will be differences in tone, albeit small.
Spruce is the most common of the two, and tends to give a brighter sound (more harmonic content, specially in the upper register), with more sustain.
Cedar tends to give a darker sound (less audible harmonics in the upper register).
Both tonewoods change with time, but at a different rate. Both become brighter with time, cedar becoming brighter faster (brighter in the tone sense). This process is known as "maturing".
The difference is small, unnoticeable for most ears, though. Here we can hear a comparison between two near identical Jose Ramirez 1A, one with 1988 spruce, and 1974 cedar for the other.
Seems that the tone qualities of both woods are so similar, that other characteristics, like strings, form, design, and luthier have much more weight in the shaping of the tone.