The white running down the roof is most likely zinc from the flashing or the galvanized cap--or even the vent pipe itself. There could also be mercaptan from the combustion of natural gas that makes up some of the white. Same with inside the furnace. Any zinc would have to be from the short connector or the cap at the top as the liner of b-vent is aluminum---not galvanized.
Mostly the white inside a furnace compartment is going to be mercaptan---the stuff they put in the gas to give it an odor---along with a little bit of zinc. You can see that same white under a pipe flashing that has nothing to do with combustion appliances---though not nearly as pronounced because there is not as much galvanized showing and no acidic exhausts to corrode the zinc.
Regardless, it is not from corrosion "inside" the vent.
- Thanks to Charles Buell for this explanation
Brown staining is simply small particles from a corroding vent being washed off the vent by rain and staining the surrounding roof-covering material.