1.) I think that Dain's chief plan is to bolster Thorin's position --
i.e. if the Men and Elves know that they are not just besieging 13 Dwarves in
the mountain, but actually having to deal with an army of Dwarves under Dain,
perhaps they will drop the siege. And, if need be, Dain will attack the
Men and Elves. If the Men and Elves shrink back from a fight, things may
have a good outcome for the Dwarves yet.
2.) The difference is the Dragon: There used to be a dragon in Erebor;
now there isn't.
3.) Roäc is more realistic than the Dwarves: the Dwarves can't grow crops
or raise animals under the mountain; they need "the friendship and good will of
the lands" about them in order to feed themselves, i.e. through trade.
Thorin, blinded by his greed, can't see this.
Even if Dain hadn't been coming, I suspect that Thorin
would have stubbornly held out until hunger forced him to
surrender.
Having talked some sense into Faramir (who had agreed to let Frodo and Sam go), the Fox was somewhat impatient to begin the journey to the Crossroads and wished that the Hobbits would hurry up.
