1) Heavy. Not as heavy, but heavy.
2) You won't be able to tell the difference. I mean it, the artwork on Dungeon Tiles has nothing on the BattleBox.
I should also say, the 4e Counter Collection that will come out in print soon also has some bonus Battle Tiles.
The Dungeon Dressings range from 1x1 squares of columns, rubble, doors, tapestries and other staples, to 2x2 squares with beds, large tables, prison cells, double doors, water, to 3x3 squares of complete inn rooms (with corridor) and stables (again, with corridor). The last two have wooden floors, but the rest have stone floor.