I was put onto this interesting point by reading Raymond Brown's commentary on Nehemiah, which I might review on here in a few weeks.
Roughly speaking, Ezra and Nehemiah were in very similar situations. Ezra, out of faith in God, did one thing. Nehemiah, out of faith in God, did exactly the opposite. And both were right.
Then I proclaimed a fast there, at the river Ahava, that we might humble ourselves before our God, to seek from him a safe journey for ourselves, our children, and all our goods. For I was ashamed to ask the king for a band of soldiers and horsemen to protect us against the enemy on our way, since we had told the king, "The hand of our God is for good on all who seek him, and the power of his wrath is against all who forsake him." So we fasted and implored our God for this, and he listened to our entreaty.
Ezra, out of faith in God, does not take a band of soldiers on his expedition from Babylonia to Jerusalem.
And I said to the king, "If it pleases the king, and if your servant has found favor in your sight, that you send me to Judah, to the city of my fathers' graves, that I may rebuild it.”... So it pleased the king to send me.
And I said to the king, "If it pleases the king, let letters be given me to the governors of the province Beyond the River, that they may let me pass through until I come to Judah... And the king granted me what I asked, for the good hand of my God was upon me... Now the king had sent with me officers of the army and horsemen.
Nehemiah, out of faith in God, asks the same king for protection on his expedition from Babylonia to Jerusalem.
So which was the right thing to do? Quite clearly, both.
Why? Because the key is acting out of faith in God, in submission to his word and after appropriate prayer (and in both cases, fasting too). And if we do that in good conscience, we're doing what is right, even if it's different to what someone else would do in the same situation.