Tumbled finish: Tiles are uneven on the edges and will most likely have holes throughout the body of the stone. This finish is made by literally tumbling the tiles with gravel in a big barrel. It is a very rustic looking tile and though they may be roughly square and roughly of uniform size, they will not have staight, nicely cut sides.
Honed finish: This is what a tile would look like if you stopped the polishing process before the tile gets shiny.
The tiles will be of uniform size and shape with square corners and straight sides and will have a silky, matte finish. Most honed tiles will be filled, so that the surface of the stone has no holes in it. These tiles are great for use in high traffic areas because they will not show wear and scratch marks as quickly as their polished counterparts.
The dust on tumbled tile comes from the finishing process and will most likely go away when you clean them. Always be sure to thouroughly clean your tiles before sealing them. Cleaning is best done with a stone specific cleaner (not a stone soap) and try not to use acid washes to get rid of grout haze.