This question concerns only the non-technical meanings of 'mode' and 'modality', and excepts the technical subject-based meanings. Etymologically, mode precedes modality.
How do they, and their Semantic Fields, differ?
a. A way or manner in which something is done or takes place; a method of proceeding in any activity, business, etc. Freq. with of. [...]
a. A particular form, manner, or variety in which some quality, phenomenon, or condition occurs or is manifested. [...]
II. In senses derived from French.
- a. A prevailing fashion, custom, practice, or style, esp. one characteristic of a particular place or period.
- a. Those aspects of a thing which relate to its mode, or manner or state of being, as distinct from its substance or identity; the non-essential aspect or attributes of a concept or entity. Also: a particular quality or attribute denoting the mode or manner of being of something. Cf. mode n. 6a.