What is the difference between cathode rays and beta rays?

cathode 'rays' were observed in electrical experiments because of their fluorescent effect near a negatively charged plate (called the cathode) in a vacuum. They were only later found to be negatively charged electrons emitted from negatively charged plates and accelerating toward positively charged ones. (Like charges repel, unlike charges attract.)

Beta 'rays' were first observed being emitted from certain unstable (radioactive) isotopes, and behaved unlike the alpha and gamma radiation also found in radioactivity. It wasn't until later that both alpha and beta 'radiation' were discovered to actually be particles; only gamma rays consist of true electromagnetic radiation. Beta 'rays' are actually electrons ejected from decaying neutrons, and are now more often referred to as Beta emission or Beta particles. 

So electrons can be described differently, not based on WHAT they are but HOW they are created or observed.

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