These words are not related, though I wish they were ... there's enough overlap with other cool etymologies I've posted earlier, as well as with compelling textilic words with wider resonances. And the Sanskrit WAR, to cover, protect, clearly seems related to veil; or to weave... The notion of concealment, covering, enclosure whether as an in-spiraling shell (vulva) or a shaggy fleece (velvet), emerges repeatedly. The need for shelter and the need to be known are intimately related to the mysteries of concealment and revelation at play in writing and textile practices.
Another element comes to mind in the practice aspect of this nexus of need and mystery: the rhythm of the practice itself, whether it be the clack of the loom (the Dogon myth of origin ascribes creation to "the creaking of the word," i.e. the sounds of the loom creaking and clacking in the weaving process) or the gentle click of the knitting needles in institutional meetings, the reassuring movement of hand and pen across paper in the kind of self-soothing relationship attributed to children's feelings for their various transitional objects, the typewriter's thump or the muted sound of the computer keyboard, or the visual rhythm of etymological dictionary entries, the lists of abbreviations, symbols, varied fonts and cases, the iterative spell of wordlists on the page and in the mind's ear.
VELVET, a cloth made from silk, with a close, shaggy pile; also made from cotton. (Ital.,—L.) 'Velvet, or velwet, Velvetus;' Prompt. Parv. Chaucer has the pl. velouëttës (four syllables), C. T. 10958; whilst Spenser has vellet, Shep. Kal., May, 185. β. Again, the form vellure occurs in Holinshed, Descr. of England, b. iii. c. 1 (R.); which is borrowed from F. velours, 'velvet,' Cot. γ. But velvet, velwet, velouet, vellet are various corruptions of O. Ital. veluto, 'veluet,' Florio; mod. Ital. velluto. The word is interesting as being almost the only Ital. word (in E.) of so early a date; it may have been imported directly from Italy. The Ital. velluto answers to a Low Lat. form villutus*, shaggy, allied to Lat. uillosus, shaggy; whilst F. velours (O.F. velous, the r being unoriginal) answers to Lat. uillosus directly.—Lat. uillus, shaggy hair, a tuft of hair; so that velvet means 'woolly' or shaggy stuff, from its nap. Allied to uellus, a fleece; orig. 'a covering' or 'protection.'—✔WAR, to cover, protect; cf. Skt. úrna, wool, lit. a covering, from vri, to cover; and see Wool. Der. velvet-y, velvet-ing.
early 15c., from O.Fr. velin "parchment made from calfskin," from vel, veel "calf" (see veal)
early 13c., from Anglo-Fr. and O.N.Fr. veil (O.Fr. voile) "a head-covering," also "a sail," from L. vela, pl. of velum "sail, curtain, covering," from PIE base *weg- "to weave." Vela was mistaken in V.L. for a fem. sing. noun. The verb (late 14c.) is from O.Fr. veler, voiller, from L. velare "to cover, veil," from velum. Figurative sense of "to conceal" (something immaterial) is recorded from 1530s. To take the veil "become a nun" is attested from early 14c.
1540s, from L. vulva, earlier volva "womb, female sexual organ," lit. "wrapper," from volvere "to turn, twist, roll, revolve," also "turn over in the mind," from PIE base *wel- "to turn, revolve," with derivatives refering to curved, enclosing objects (cf. Skt. valate "turns round," ulvam "womb, vulva;" Lith. valtis "twine, net," apvalus "round;" O.C.S. valiti "roll, welter," vluna "wave;" Gk. eluo "wind, wrap," helix "spiral object," eilein "to turn, squeeze;" Goth. walwjan "to roll;" O.E. wealwian "roll," weoloc "whelk, spiral-shelled mollusk;" O.H.G. walzan "to roll, waltz;" O.Ir. fulumain "rolling;" Welsh olwyn "wheel")