There's an open access article "From vitamin D to hormone D", which explains the difference(s). Vitamin D is a family of substances, but for humans the most important one from this family is vitamin D3 aka cholecalciferol. "Hormone D" (1α,25(OH)2D3, the substance that normally regulates calcium etc.) aka calcitriol is closest to vitamin D3. So if one wants to be technical about it, the vitamin(s) and the hormone are slightly different substances. Vitamin D itself is more aptly called a prohormone (i.e. a hormone precursor):
Vitamin D is not technically a vitamin, ie, it is not an essential dietary factor; rather, it is a prohormone produced photochemically in the skin from 7-dehydrocholesterol. The molecular structure of vitamin D is closely allied to that of classic steroid hormones (eg, estradiol, cortisol, and aldosterone) in that they have the same root cyclopentanoperhydrophenanthrene ring structure. Technically, vitamin D is a secosteroid because one of the rings of its cyclopentanoperhydrophenanthrene structure has a broken carbon-carbon bond.
The above view is also endorsed by Horst et al., but they have a much more detailed discussion, including the fact that in vitamin D poisoning, calcifediol (25(OH)D3)--an intermediate compound in the pathway between the vitamin and the hormone--can itself act as a hormone, i.e. it has some binding affinity to the calcitriol receptor, but only at at very high doses.
As for how the vitamin label came about, "historical accident" is one way to put it...
Considering the fact that now we accept that the biologically active form of vitamin D [what the other paper calls "hormone D"], namely 1a,25(OH)2-vitamin D3, is a steroid hormone, it is somewhat ironic that vitamin D, through a historical accident, became classified as a vitamin. It was in 1919/20 that Sir Edward Mellanby, working with dogs raised exclusively indoors (in the absence of sunlight or ultraviolet light), devised a diet that allowed him to unequivocally establish that the bone disease, rickets was caused by a deficiency of a trace component present in the diet. In 1921 he wrote, "The action of fats in rickets is due to a vitamin or accessory food factor which they contain, probably identical with the fat-soluble vitamin." Furthermore, he established that cod liver oil was an excellent antirachitic agent.
Shortly thereafter E.V. McCollum and associates observed that by bubbling oxygen through a preparation of the "fat-soluble vitamin" they were able to distinguish between vitamin A (which was inactivated) and vitamin D (which retained activity). In 1923 Goldblatt and Soames clearly identified that when a precursor of vitamin D in the skin (7-dehydrocholesterol) was irradiated with sunlight or ultraviolet light, a substance equivalent to the fat-soluble vitamin was produced. Hess and Weinstock confirmed the dictum that "light equals vitamin D". They excised a small portion of skin, irradiated it with ultraviolet light, and then fed it to groups of rachitic rats. The skin that had been irradiated provided an absolute protection against rickets, whereas the unirradiated skin provided no protection whatsoever; clearly, these animals were able to produce by uv irradiation adequate quantities of "the fat-soluble vitamin", suggesting that it was not an essential dietary trace constituent. In parallel studies, Steenbock and Black at the Biochemistry Department of the University of Wisconsin found that rat food which was irradiated with ultra violet light also acquired the property of being antirachitic. However, because of the rapid rise of the science of nutrition -- and the discovery of the families of water-soluble and fat-soluble vitamins -- it rapidly became firmly established that the antirachitic factor was to be classified as a vitamin.
As an amusing parallel to this, calcifediol was identified in 1968 as an active metabolite of the vitamin, but it wasn't until 1971 that calcitriol was properly identified as the metabolite that exercises the hormonal action under normal circumstances. (cf. DeLuca)