"Run away" and "run along" may seem similar but are used quite differently. The difference is idiomatic. Your question title says one but the body of your question asks about the other, so I will address both.
Run away is self explanatory. It means to depart (at speed) from something, because "away" is a relative direction meaning anywhere but here.
Run along is an idiom which is an order to leave, not in any particular direction, but "along" implies that the person continue with their other business.
"run along now, there's a good girl"
I would describe this as quaint British English and would expect to see it in period literature, either said to a little girl by an elder as a way of asking her to perhaps go and play elsewhere; or perhaps said to someone older as way of patronising them. The implication is that they are behaving, or being a "good girl" by following the order to go away.
In this instance "there" is not used as a direction, but as a sort of introduction, the way you might say "there's somebody at the door".