What is a Oscilloscope?
An oscilloscope with long persistence has a screen coated with a special type of phosphor (the material which glows when bombarded with electrons) which will continue to glow at the spot the electrons have hit for a few minutes after the electrons have been removed.
It is sometimes necessary to compare two traces. This is most easily done using either signal and then switches to trace out another signal. The long-persistence screen of this type of oscilloscope displays both traces at the same time.
Dual-beam oscilloscopes, on the other hand, use two electron beams. Each beam traces out a different signal. Both signals are displayed simultaneously on the screen. A long-persistence screen is not necessary in a dual-beam oscilloscope.