Assigning Roman numerals to chords (harmonic movement) is just a form of notational shorthand to make processing harmonic information quicker and easier. It grew out of a 17th and 18th century practice called figured bass. Roman numerals denoted the chord to be played above a specific bass note and Arabic numerals attached to them spelled out inversions and melodic anomalies. For example V4/2 tells me that the 7th is the lowest voice in the dominant 7th chord and will likely resolve down a half step. Similarly II6-5 tells me that the 6 note is an appoggiatura and should likely be accented. It's hard to imagine how one would navigate the harmonic terrain of say a Sor study or the Aranjuez Concerto or even make coherent guitar arrangements of Turlough O'Carolan pieces without at least a basic harmonic analysis underpinning the effort.
Beyond that an understanding of functional harmony contributes to interpreting pieces more musically. It directly impacts phrasing, voice-leading, accents, timbral decisions, etc, etc, etc. I'll be the first to concede there's little or no value to knowing harmony in several currently ubiquitous pop idioms but for classical, jazz, and a great deal of folk music it's incredibly useful, even essential, IMHO.
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