One of the most important things that any one ever said to me about reading Shakespeare is this: "Plays are written to be performed." That single statement completely changed my philosophy on teaching. It helped me overcome my fear of teaching Shakespeare and set me on a path to using Shakespeare as much as I could in the classroom.
It's very difficult to say exactly what would be different, but I can give an example of how seeing Hamlet performed would be different than reading it. Let's look at Hamlet's famous "To be or not to be" soliloquy - perhaps the most famous speech in the English language. This part of the soliloquy comes in the middle. There are phrases and words that might be difficult to understand when sitting and reading it. But when you watch someone perform it, it comes alive:
Who would bear the whips and scorns of time,
The oppressor's wrong, the proud man's contumely,
the pangs of despised love, the law's delay,
the insolence of office and the spurns
that patient merit of the unworthy takes
when he himself might his quietus make
with a bare bodkin?
All of these words might merely be words to the reader, but to the audience member, they come alive. An audience member might not be able to understand the words. He or she might not know what a contumely is or what a bodkin is. An audience member doesn't necessarily need to know the definition of these words. The actors and their interpretation make the audience member feel what it means by showing them. This is the essential difference between merely reading the play and participating in it as an audience member.