What is the difference between fuel oil and diesel fuel ?

Because they are different molecules.  When you have unrefined oil and diesel, they are all combined.  Diesel, oil, kerosene, tar, and others are all mixed into one pile called petroleum.  Diesel is one strand of carbons and oxygens and oil is another.  They burn differently and require different engine mechanisms to make them ignite.  If you put unleaded fuel in a diesel engine, it would make the engine useless unless the fuel was emptied.  The same goes for diesel fuel in an unleaded engine.  

It is odd that your heater says that it will take both unless it has some way of igniting both without the chance of harming the engine.  I'm not sure how that works, but if it says that it can do it, believe them.

I agree with the other answerer, but you should note that just because it says it can burn multiple fuels, it most likely requires different accessories to do so.

Fuel oil is a lighter type of oil, a liquid byproduct of crude oil used for energy, especially heating. When petroleum is refined, there are two main categories under which it is classified. One is distillate oils, which includes diesel, and the other class is residual oils, which includes things like kerosene. Different types of fuel oil are classified under both categories. Distillate fuel oil is the type generally used for home heating.  Gasoline is also a product of petroleum refining, but it's less stable than the fuel oil used to heat homes and commercial properties.

So you can burn diesel fuel *and* fuel oil because diesel fuel is just a type of fuel oil.  You have many different flavors of Kool-Aid, but they're all basically the same thing, as far as your body considers using their calories.

Perhaps do you mean the difference between #2 heating oil (which is what most home oil burners use) and diesel fuel (like what trucks and diesel cars use). Technically, it is the same stuff, the main difference is the tax ! If it is used for vehicle fuel there is a higher excise tax (they also dye the two materials 2 different colors so you can tell heating fuel from vehicle fuel). There are a lot of people who fill their diesel cars from their home oil burner tank to save money on the fuel tax; it is one of those things that is technically illegal, but you never hear about anybody getting arrested for it. I think kerosene is a slightly different cut from the petroleum refining process. Usually kerosene is de-odorized so, if this is a heater where the actual device is taken inside, you might want to stick with the kerosene, because the heating oil and diesel both smell pretty nasty, it will even make some people puke !