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Please explain for me the difference between chromatic root movement and modulation in the chord progression of a song?

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Chromatic root movement would typically refer to a progression of chords--all in the same key signature--where the root descends chromatically or ascends chromatically. For example, this can be achieved using a tritone substitution:

| Dmin | Db7  | CMaj |

The root descends chromatically from D to Db to C. The entire progression is in the key of C, and the chromatic movement doesn't imply a change of key/tonal center. As a more dramatic example, we could have something like this:

| CMaj | GMaj/B | BbMaj | FMaj/A | E7alt/G# | CMaj/G | D7/F# | G7 | 

Again, this entire progression is centered around the key of CMaj. While there may be some passing chords that don't fit perfectly in CMaj, nothing here qualifies as a key change to a different tonal center.

By contrast, modulation implies moving from a first key to a second, different key. For example, the song could modulate from a tonal center of CMaj to DbMaj:

| CMaj | FMaj | G7 | Ab7 | DbMaj | GbMaj | Ab7 | GbMaj | DbMaj |

In the third and fourth bars, there is ascending chromatic movement in the bass, and this is due to a key change from CMaj to DbMaj.

So in short: chromatic root movement typically implies chromatic ascension/descension in a single key, whereas modulation implies moving to a new key. Both can involve movement along the chromatic scale.

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