What? Isn't that the same thing?

You're basically saying:
[I'll go] after you
or
[you go] before me

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[You are] right.
But it's far more natural to eliminate [you] than [I] in English. Every imperative command is like that, "[you] go to the store", "[you] go before me". It's not such a stretch to "[you go] before me". But it seems bizarre (apart from the fact that it is idiomatic) to lop off "I'll go". You can't just say, "To the store."

It also seems strange that the message A wants communicate to B is "you (B) should go first", but it is done by referring to what A is going to do afterwards.

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