I often see roman numerals and figured bass together when labelling chords but are both really necessary? Surely the Figured bass will have accidentals if the chord is to be modified in some way so doesnt that make the chord label redundant?
The Roman numeral label describes the function of the chord; the figures describe the inversion or other modifications. Figures by themselves only describe which notes to play, and Roman numerals by themselves only describe the basic function of the chord. The two together give a more complete description of the specific way in which the notes of the chord are expected to behave within a piece of music.
Figures are used with Roman numerals to show the inversion. In their original function as performance shorthand, the figures accompany a bass part, so they indicate which chord to play. For example, given a C bass note and a key signature of one sharp, a 6 means the chord is A minor while the absence of a figure means it's C major. If the key is G major, that's either ii6 or IV.
If you know any three of four quantities, you can determine the fourth:
- bass note
- Roman numeral
The figures, in other words, aren't strictly necessary when you write a roman numeral analysis under a score, but the figures allow the analysis to stand alone. Without the figures, for example (or, more precisely, using only 7 to indicate 7th chords without regard to inversion), it wouldn't be clear that this is a harmonization of an ascending scale:
I V4/3 I6 ii6/5 V IV6 V6/5 I