Why is balsa wood classified as hard wood when it is very soft and light? What is the difference between hard and soft wood?

The terms hardwood and softwood don't relate to the weight or density of the wood, but to the tree type. Hardwood trees are angiosperms (mostly decidous in the northern hemisphere but evergreens in the southern hemisphere), while softwoods are conifers. There are many more types of hardwood trees than there are softwood.

https://en.wikipedia.org/wiki/Hardwood

The distinction between hard wood and soft wood is not in how hard and soft the wood is (that's too obvious); but, in how their seeds are formed in the trees. Softwoods, suchs as pines, spruce, and fir, are examples of gymnosperms, plants that produce seeds without covering. Hardwoods are a type of angiosperm, a true flowering plant that bears seeds enclosed in capsules, fruit, or husks. Hardwoods also tend to lose their leaves in temperate climates, whereas softwoods are evergreens; but in tropical climates many hardwoods retain their leaves.

https://en.wikipedia.org/wiki/Hardwood
https://en.wikipedia.org/wiki/Softwood