Is it True that Propane Exists as a Liquid and a Vapor?

Yes!  Both exist inside your propane tank.

Liquid propane and vapor cannot be used interchangeably.  A propane system designed to use vapor propane can't utilize propane in it's liquid form. 

Liquid propane use is generally limited to industrial and agricultural applications.  For example a farmer may use liquid burners to produce extremely hot temperatures to destroy unwanted seeds and weeds in his field.  The liquid propane travels through the coils seen in this picture.  As the coils heat up they aid in the rapid conversion of liquid propane to vapor gas.  Without this process the liquid propane would exit the burner orifice and begin to freeze the equipment since liquid propane flashing to a vapor gas can become 44 degrees Fahrenheit below zero.

Smaller vapor burners are also available.  The Manchester Power Jet Hand Torch or as some prefer to call it, the Manchester Power Jet Weed Burner is a hand held burner used for numerous applications.  Many times you will see city workers using these torches as they fill pot holes in the road. Here is a picture on one.  These torches are available in three foot and 8 foot lengths.