Scatter and shatter are the same word, split by the boundary between Norse and native English. Old English softened its sc- to sh-, so the native form of this word is shatter; the sk- form scatter is a northern English variant reflecting Norse influence, where no such softening happened.
That is the single most useful test for spotting Scandinavian influence in English: where you find sk- you are usually looking at a Norse word or a Norse-influenced form, because the native line would have given sh-. Ship and skipper, shirt and skirt are the same pair. Scatter and shatter then went their separate ways in meaning, scatter taking the sense of dispersing and shatter the sense of breaking, which is how English usually resolves having two forms of one word.
scatter (v.) mid-12c., scateren , transitive, "to squander;" c. 1300, "to separate and drive off in disorder;" late 14c., "to throw loosely about, strew here and there," possibly a northern English variant of Middle English schateren (see shatter ), reflecting Norse influence. The intransitive sense, "go or flee in different directions, disperse" is from c. 1300. As a noun from 1640s, "act or action of scattering;" by 1950 in reference to radio waves. also from mid-12c.