TL;DR (Actual Question:) I'm wildered; so please explain as though I were 10 years old.
What are the similarities and differences? This doesn`t compare all 4 nouns simultaneously.
A Student's Introduction to English Grammar (2005; by Huddleston and Pullum) uses all 4 terms (3 concurrently, in the penultimate para of page 53). But the book never defines each. OED's definitions (below) are uselessly circular. ♦ Because OED's definition 2 for the noun 'modal' just links to the definition of the ADJECTIVE 'modal', I quote the latter.
♦ modal {adj} = 5. a. Of or relating to the mood of a verb. Of a verb or other element: expressing or used to express modality.
modality = 5. Grammar. The quality or fact of being modal (see modal adj. 5).
mode = 2. a. Grammar. = mood n.2 1.
[ODO] mood2 = 1. Grammar. a. A category or form which indicates whether a verb expresses fact (indicative mood), command (imperative mood), question (interrogative mood), wish (optative mood), or conditionality (subjunctive mood). (cp OED mood n.2).