Does a script usually have to come out like a play? Or, there should be a difference?
A "script" is general; a "screenplay" is something to be filmed. A "play" is something to be performed on a stage, and more limited because of that; for example all kinds of things can be done in a screenplay, with special effects, sound effects, dubbing, green screens, etc, things you just can't do on stage. For example, a person can literally dissolve or explode, or have their head blown off.
So while the screenplay and a stage play are very similar, and the acting is similar, they have slightly different conventions for what is included. Read Screenplay, Stage play, Teleplay - What's the difference? as a starting point.
There are technical differences between a play and a screenplay. Amadeus has set you on a good path.
The, often overlooked, important difference between a play and screenplay is the required location discipline. A director may move cameras at any point to any location but there is only ONE stage.
Subsequently, if you were writing a play: "Emigrating to the New World."
ACT ONE: In the current family home.
ACT TWO: Aboard the ship for the long journey.
ACT THREE: In their shiny new home.
The above play would require TWO scene changes. You cannot switch back and forth between locations.