Cane sugar is a product of a tall grass found in a tall, tough bamboo type grass called sugar cane. Sugar cane grows in warm, humid climates such as Louisiana, Hawaii, and some Central American countries. The sap of the sugar cane is collected after harvest. The water content is evaporated away leaving behind s sugar based syrup which can be further evaporated to form a rough sugar crystal, or raw sugar. Raw sugar can further be refined producing a white, pure crystal of sucrose.
Regardless of the company that goes through this process, the end product of refined sugar cane sap is sucrose. The sucrose prepared this way is over 99% pure sucrose with very little impurities. It may be possible that some companies might "cut" the sugar with some other food grade sweetener in which case it may be more or less "sweet". However, this product could not be sold as pure sugar, but would have to be marketed as some other kind of sweetener.
It is worth noting that sucrose can be obtained from other sources besides sugar cane such as sugar beats. However, regardless of the original plant source, the final product is still sucrose.