Cane and beet sugar are both sucrose and are chemically the same. Unrefined brown sugar can still be found, but it is imported these days and usually costs more than domestic brown sugar. Domestic brown sugar is really refined sugar with molasses added. It will affect both taste and color of the wine, but for some wines it is required. Corn sugar is dextrose, preferred for beermaking but tradionally avoided by winemakers.
Honey is another subject altogether.
It comes in many, many flavors, depending upon the flowers the bees predominately visited while collecting pollens and nectares used to make it. These flavors do affect the wine, but so does the honey itself. Honey tends to mellow out a wine and contributes ever so slightly to body. Some people prefer it for that reason alone, while others prefer it for ecological reasons. My problem with honey is that it slows down the clarification process considerably.