Despite what Wikipedia says, it's not always 100% reliable as a source of information. Garden wire is NOT garden twist wire; the term garden wire may refer to large rolls of chicken wire or clematis mesh. Other types come in a roll or circular shape, may be green plastic coated or just plain wire, and this is used for various purposes, depending on its thickness. Those purposes include stringing horizontally between hooks in a wall or fence, tightly, to provide something to tie climbing plants to as a support; I've used a heavier duty one to fix hanging bird feeders to railings, or hanging lights. It is not usually used to tie plants to anything because it's likely to damage plant stems.
Garden twist wire, on the other hand, is much, much thinner wire, rather like fuse wire, and is sheathed in broader, softer plastic which is much less likely to damage plant stems, and can be twisted to provide a light support, such as supporting a plant stem to a bamboo cane or stick. Even so, it should never be twisted in such a way that the plant stem is compressed by it; it should be wound twice round the cane and only once round the stem, and the tight twisting should be on the supporting cane or stick rather than the plant stem.