Cane juice is the juice pressed out of a stalk of sugar cane. Evaporated cane juice is the brown crystalline sugar from sugar cane juice that has been allowed to dry. In most of the world, cane juice and evaporated cane juice are the most natural forms of sugar.
In Brazil, you can find street vendors hawking caldo de cana, a mixture of sugar cane juice and lemon. In Vietnam, one of the most popular of all beverages is nước mía, which is a mixture of sugar cane juice and kumquat juice. In Pakistan there is roh and in India there is ras, and Egypt you can find aseer asab, all refreshing beverages that may or may not be mixed with a little tart fruit juice and served over ice.
In the United States, more and more manufacturers of bottled fruit and even some vegetable juices are adding small amounts of cane juice and evaporated cane juice as a natural sweetener. In the USA, however, this natural sweetener can be subjected to a process similar to the way high-fructose corn syrup is made to change the glucose and fructose content of the cane juice to make it sweeter. This doesn't have to be declared on the label, although there are companies that use "Sugar in the Raw" who simply add more sugar to make a juice sweeter rather than modifying glucose and fructose content.
Is cane juice healthier than white sugar? Yes, but in a very specialized ways. There are complex carbohydrates in cane juice that are refined out of white sugar that stimulate the immune system. One of them fights a parasitic infection known as coccidiosis (not to be confused with the fungal disease that occurs in the Phoenix area in Arizona in the USA, coccidioidomycosis). Probiotic Lactobacillus bacteria thrive on sugar cane juice. Sugar cane that has had to fight off an infestation of insects known as sugar cane nymphs develops extra antioxidants (and you'll be happy to know that most American cane juice producers manage to get the bugs off the sugar cane before it goes into the juice press). Because cane juice is treated with lime (the kind of lime in limestone, not the kind of lime that grows on a tree), it's not a bad source of calcium.
But cane juice and evaporated cane juice are still sugar. They have a little more nutrition than white sugar, but if you need to avoid white sugar, you need to avoid cane juice and evaporated cane juice, too.