How does Shakespeare's theater differ from a modern day theater?

I think you are asking about the physical theaters in which Shakespearean plays were first performed, and how these might compare to the theaters in which we'd expect to see the plays now. Shakespeare's plays are most associated with the Globe Theatre in London, which was actually the first purpose-built playhouse of its kind and was financed by Shakespeare's acting troupe.
Prior to the building of The Globe, plays were regularly performed in any space that was large enough to host them, be it a village square, a pub, a schoolhouse, or a royal court. The Globe, then, has far more in common with our modern theaters than it did with the "theaters" of the time before it. Still, it differed somewhat from the modern understanding of a theater.
In the first place, it had no roof, being open to the air and shaped like a Roman amphitheatre. Only the stage had a roof ("the heavens") for the purposes of hanging scenery from it. People would sit all around the circular theater on the raised banks of seats, but there was also a "pit" in front of the stage where people could stand, for a smaller fee than a ticket with a seat. There was also a balcony above the stage, probably for musicians.
https://www.shakespearesglobe.com/discover