What is the purpose of interrupts? How does an interrupt differ from a trap? Can traps be generated intentionally by a user program? If so, for what purpose?

The purpose of the interrupt is to cause the computer or processor to work on a different task temporarily and after finishing that work return to its previous work.

The difference between interrupts and traps is that the interrupts are hardware generated interrupts while the traps are software generated interrupts.

Yes, user programs can generate traps intentionally. So that it can be used to call operating system routines or it can arithmetic errors.