discard implies the letting go or throwing away of something that has become useless or superfluous though often not intrinsically valueless.
discard old clothes
cast, especially when used with off, away, or out, implies a forceful rejection or repudiation.
cast off her friends
shed and slough imply a throwing off of something both useless and encumbering and often suggest a consequent renewal of vitality or luster.
shed a bad habit
finally sloughed off the depression
scrap and junk imply throwing away or breaking up as worthless in existent form.
scrap all the old ways
would junk our educational system
Word History
Etymology
Verb (1)
Middle English sheden, shedden, shoden "to divide, separate, part or comb (hair), pour out, discharge, cause (blood) to flow, pour (tears) in drops, emit (light)," going back to Old English sceādan, scādan (Class VII strong verb) "to separate, divide, distinguish, decide, scatter, shed (tears, blood)," going back to Germanic *skaiđan-/skaiþan- (whence also Old Frisian skētha, skēda "to separate, depart, exclude, decide, distinguish", Old Saxon skēdan, skēthan "to separate, render (fat), disperse," Old High German skeidan "to separate, distinguish, apportion," Gothic skaidan "to separate"), going back to pre-Germanic *skoit-, variant (with o-ablaut and voiceless final stop) of Indo-European *skei̯d- "split, separate," whence also, with zero grade, Greek schízein "to split, separate," with zero grade and nasal present Sanskrit (Vedic) chinátti "(s/he) cuts off, tears up," Latin scindō, scindere "to split, cleave, separate," with o-grade Russian Church Slavic cěditi "to strain, filter," Bosnian-Croatian-Serbian cijèditi, Lithuanian skáidau, skáidyti "to separate, break down" (also, with e-grade, skíedžiu, skíesti "to make thin, separate")
Note: Old English sceādan is a strong verb in West Saxon, but already a weak verb in Northumbrian, and predominantly a weak verb in Middle English. The Middle and Modern English outcomes of the verb, excepting the uncommon form shoden, seem to reflect Old English scēadan, with later shortening of the vowel, as in other monosyllabic words ending in -d (see note at red entry 1). — The Germanic outcome of the Indo-European base, with o-grade and t, is peculiar; Lexikon der indogermanischen Verben, 2nd edition (Wiesbaden, 2001), suggests generalization of o-grade and devoiced d from a presumed reduplicated present 3rd-person singular ending in -ti. The base *skei̯d- likely emerges directly in Germanic as *skītan- "to defecate, shit" (see shit entry 2); this sense may have co-opted the verb, leading to adoption of the variant *skaidan- to express the other, less objectionable meanings. The Lexikon writes the Indo-European etymon as *sḱhei̯d- to account for the Indo-Aryan forms and the aspiration in Greek, though Slavic and Baltic do not show a palatovelar. R. Beekes (Etymological Dictionary of Greek, Brill, 2013) regards the Greek aspiration as unexplained. Following the notion that -t-/-d- is a "root extension," J. Pokorny (Indogermanisches etymologisches Wörterbuch) adduces a more basic root *skei̯- "split," which, with other root extensions, produce for him a vast range of vocabulary in Indo-European languages.
Noun (1)
Middle English shed, shode "part in the hair, difference, distinction, faculty of discernment," going back to Old English sceād, gesceād "separation, distinction, understanding, reason, reckoning," going back to Germanic *skaiđ- (whence also Old Frisian skēd "part in the hair, crown of the head," Old Saxon skēd, skēth "distinction," Old High German gisceid "knowledge, reason, distinction"), nominal derivative, with varying stem-formatives, from *skaiđan-/skaiþan- "to separate, distinguish"; (sense 1) derivative of shed entry 1 — more at shed entry 1
Note: As is the case with shed entry 1, the Middle and Modern English outcome of Old English sceād appears to reflect a variant scēad, with later shortening of the vowel. The meaning "a divide of land" (sense 2), attested earlier in watershed (18th century) appears to be an outgrowth of earlier physical senses of the word, as "the part of one's hair" or "a passageway between the threads of a warp on a loom," though its very late appearance relative to the other senses makes the relationship uncertain.
Noun (2)
Middle English shedde "lean-to, penthouse," perhaps from a Kentish variant of shudde "hut, lean-to," going back to Old English scydd "swine pasture," of uncertain origin
Note: The word shudde in the Promptorium parvulorum, a 15th-century English-Latin dictionary, is used synonymously with "lytylle howse," "to-falle" [lean-to], and "hovel, or swyne kote, or howse of sympyl hyllynge [shelter] to kepe yn beestys." The approximate synonym "swyne kote" (see swine, cote entry 1) would seem to connect the word with Old English scydd, a word attested in charters that is taken to mean "swine pasture" (the general sense then perhaps being "place to keep swine"). The incidences of this in place and personal names are discussed at the entry for Gunshot Common in the Survey of English Place-Names volumes for Sussex, edited by Allen Mawer, F.M. Stenton and J.E.B. Gower (see Survey of English Place-Names on line). The editors note that the expected Old English source would be schudd rather than shydd if shudde is the regular outcome in East Anglia (presumed to be the source of Promptorium parvulorum?), though most of the onomastic examples in the Middle English Dictionary are from Sussex, Surrey or Kent. The Oxford English Dictionary's hypothesis that shed is somehow a variant of shade (see shade entry 1) is not persuasive, despite the forms shadde in Caxton (1481) and shaddys in Richard Arnold's chronicle (1503?).