Wherever "let go of your hand" is used, can "let go your hand" be used in its place? Is there any difference at all?
"Let go your hand" is grammatically correct, but archaic. See Shakespeare's Julius Caesar: "let slip the dogs of war."
No, you shouldn't drop the prepostion of in the expression let go of.
The expression let go is used for ceasing employment, so the result could easily become confusing or misleading.
There is nothing grammatically incorrect in your phrase.
Let your hand go.
It is the same, where "your hand" is the direct object of the verb "let". You've simply inverted part of the sentence, which has no bearing on the grammatical soundness thereof.
It does sound a bit odd, all the same, at least, to me (native US English speaker) it does.
In Spanish we say "letting go of hands" or literally "soltarse de las manos". I think in English it would be like "hands off" or "release your hands" but I don't know if it replaces "let go your hand".
As other stated - grammatically correct, but bit odd and even ambiguous, unclear.
Therefore the meaning is not the same:
Let your hand go
is different from
Let go of your hand
(which does imply it was held, where the first example does not)
If it were "Let go my hand", I would have no hesitation in accepting it as a spoken alternative to "Let go of my hand".
But "Let go your hand" is odd, because "Let go of your hand" is an unlikely thing to say, so I would probably interpret it as "Let go (of something) with your hand".
In context of the whole it is perfectly understandable, and therefore correct and acceptable English; writing gospel song lyrics and poetry I find it is literary and beautiful.
In modern, informal (usually rapid) speech, a constituent can be placed at the end of a sentence for focus. The constituent can be one that's not actually licensed by the argument structure of the clause.
In the OP's example (archaic reading aside), your hand is not actually licensed. The phrasal verb let go takes a subject, and possibly a for- prepositional phrase as a complement. It does not take a direct object. But you add your hand for assertive focus. This construction is grammatical, but only in informal speech (including a written account of a conversation), not in formal writing.
Another example: Turn it down that noise. Here that noise is normally not licensed because it already fills the object position, but it gets tacked on to the end of the utterance for focus.