The short and easy answer to the question of how cancer cells differ from normal cells in the body (normal as in healthy) is that a cancer cell is a skewed version of a healthy cell. The primary hallmark of cancer at the cellular level is of mass, unceasing proliferation.
Healthy cells know when to take the steps needed to mature. This is a process known as differentiation. They also know when to die—called apoptosis.
A cancer cell is a cell that was born of a healthy cell but something went wrong in the delivery. Instead of being an exact image of the parent cell, it has a slight tweak—a chromosome didn't make the transition properly, for example. Normally, this mutation is enough either to cause the cell to die or to cause it to be killed by the immune system. This kind of cellular miscue happens with greater frequency as we age.
On rare occasions, the outcome isn't quite so benign. Sometimes the mutation leads to the formation of a new gene. If this gene gets switched on by a certain enzyme, there may be no way for the cell to shut itself down through apoptosis, nor a way for the cell to differentiate—for the cell to correct itself. As a consequence, the cell continues to survive in its skewed state, past the time when it should have matured and died.
So we now have a living cell that has lost its ability to die and is proliferating, or creating daughter cells made in its image. As more and more of these cells begin to appear in the body, they will soon begin to cause problems in the human body, leading a person to become symptomatic.
For instance, let's say that this out-of-control cell is part of one's lung. As the cells divide and double, they begin to occupy space in the body. Because the occupation is a relatively slow one, the body adapts to this tumor formation. As long as the body can adapt to it, there won't likely be many symptoms. But if it gets big enough, it will run out of space and begin to hit things like nerve endings. The ensuing pain might bring a person to the doctor. Or the tumor may be large enough to occupy enough lung capacity to make a person get fatigued very quickly.
To say that cancer is the result of a single mutation is to simplify the disease, because that mutation requires other things to go wrong in order for it to survive. However, once it gets going, that original mutation will be passed to other cells in the tumor. Along the way, more mutations occur.
As the mutations and changes build up, the cancer becomes more and more difficult to treat. This is chiefly why an early-stage cancer is more treatable than a late-stage cancer; in the early stages there are far fewer of those mutations that separate a cancer cell from a healthy cell.
Finally, all cells—healthy and cancerous—process sugar to survive. Cancer cells, however, consume sugar at a slightly higher clip. This quirk has allowed for the development of some imaging techniques that let doctors see more clearly whether treatments are working or not, or to what extent a cancer has spread.
Uncovering the differences between a cancerous cell and a healthy cell does advance medicine, but the better revelation would be to uncover the difference between a cancer-causing substance and a benign substance. After all, it's better to prevent cancer as best you can than to have to treat it after it has developed.