Writing an interpreter can be a lot simpler than coding an equivalent compiler. Suppose you have an old game you want to play that was written in hand crafted 8080 ASM. You can emulate an old CPU at reasonable speeds and get a complete working solution. But, if uses say self modifying code there really is no way to directly map 8080 to x86(64) instructions.

But, a more common case is it let's you debug systems. Even if you can't run an iPhone app in full speed on your desktop you can inspect what's going on in detain with an emulator in a way you can't really do with compiled code.