In general "demonstrate" will mean someone will physically show you how to do something. Prove may be done by talking, or writing something down, or giving you an internet link and so on, and doesn't necessarily involve them showing you how to do it. Sometimes however in conversation they will be used interchangeably.

Prove- You will explain the truth about something through actions or words. We usually use "prove" in conversation. Example: "I will prove how much I care about you by cooking dinner for you." Demonstrate- You will show the truth through physical action. Example: "I will demonstrate this experiment in class today."

prove = give evidence demonstrate = show off