Why does my new smart dimmer have an extra wire?
Your old dimmer had neutral, eh? Now think that through.

Let's see. We've got safety ground; then always-hot as supply; then neutral to power the device -- Wait, aren't we missing something? How does it power the light??? 

The answer is, that wire you have identified as neutral, is not neutral at all.  It is surely switched/dimmed hot to the light.  The old dimmer does not need neutral to power itself; it's designed to get power by leaking it through an incandescent bulb.  This is the old way of doing things, and it does not play well with LED.  

Now, I seriously doubt the old dimmer had a white output wire.   What I suspect is the wire in the wall is white.     That's bad news: it means your wiring the is an old-style switch loop which is using white as a hot wire.  That style of wiring does not have neutral.  You cannot use that smart switch here, unless you want to pull a /3 cable into the wall and wire it as a new-style switch loop.