Why didn't they use the same pass?

[Manwe Sulimo]

I always wondered about that, too.

When you think about it, it would have been much easier, passing over the Misty Mountains in a protected (by Rivendell) area, traveling through the Beronings' lands, down Anduin (the water keeping the Nazgûl at bay), to Lothlórien, then continuing to Mordor. They would have avoided Moria (and the Balrog), Isengard, and been protected from the Nazgûl at (almost) every turn.

The only things I can find wrong with that path are these:

1) Mount Gundabad: The "capital" of the Misty Mountain orcs, anybody trying to pass through would have probably run into an orc or two (thousand)....remember, in The Hobbit, Thorin & Co. were captured and almost killed.
2) Gladden Fields: Both Sauron and Saruman had been keeping an eye (or the Eye) on the Gladden Fields, where the Ring passed out of knowledge 3000 years before.
3) Dol Guldur: Passing down Anduin or west Mirkwood would have brought the Fellowship too close to Dol Guldur, and all them trolls, orcs, and Nazgûl (Anduin couldn't protect them, then) would certainly capture or kill all the Fellowship, even if there was support from Lórien.

And also remember: any trip that way would result in the destruction of Helm's Deep, Edoras, and (probably) Minas Tirith.