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Your nose is a bit better at humidifying the air before it hits your lungs, so breathing in through your nose makes your throat less likely to dry out.

But some people's noses just don't have the throughout to support running (mine are this way), so you can basically disregard the advice.

Yeh, what @rrrrrrrrrrrryan said, I think it's no great problem either way :-)

If you're only breathing through your nose you're still walking.