If you and I each saw "Hamlet" or "A Midsummer Nights Dream" around the same time, but performed in two different locations by two different theatrical companies, would there be significant deviations in the performances we saw (that is to say, beyond just minor differences in the way particular actors delivered their lines/their overall acting ability)?
We know that comic actors were improvising on the stages of London in 1589 - 1613 because Shakespeare asks them to stop doing it:
...
And let those that play your clowns speak no more than is set down for them- for there be of them that will themselves laugh, to set on some quantity of barren spectators to laugh too, though in the meantime some necessary question of the play be then to be considered. That’s villainous, and shows a most pitiful ambition in the fool that uses it.
There were also "unofficial" copies of Shakespeare's scripts in circulation at the time. It is possible (though by no means certain) that you could have seen two very different versions of Hamlet in two different locations. The First Quarto of Hamlet (or Q1) was published in 1603. It is roughly 1600 verse lines shorter than the other two contemporary versions (F1 & Q2) but is often similar in plot and language.
One of many theories about this so-called "bad quarto" is that it was a reduced version of Shakespeare's play intended for use by a touring company. If that is true a person could have seen two versions of Hamlet in two locations. Take this all with a grain of salt. The exact purpose of Q1 and its place in the canon are still a matter of open debate.