The main difference is that Nathan Lane was much funnier than Jason Alexander. There's a point in the movie where Alexander realizes that it's ok for the character to camp it up and then he takes off. Lane camped it up from the second he set foot onstage and hit every conceivable laugh in the script.
"Is it a one-set show?"
It was a bare stage and a lot was left up to the imagination.They slid things on for indoor scenes (chairs, etc), they used a trap door for a canoe, they used a special lighting for hiding in the closet, etc.
"How was the twin thing handled -- were they ever onstage together, i.e. using a double with his back to the audience or something lame like that?"
Mostly it was handled through costume changes, but they were onstage together one time when the mean twin confronts the good twin. In that moment, the actor was basically talking to an empty wing-back chair. The chair was facing upstage with its back to the audience.
"Is there nudity in the stage version"
The only reason it sold so many tickets and ran as long as it did on Broadway was because at some point every actor was nude.
If anyone ever tells you that you put too much Parmesan cheese on your pasta, stop talking to them. You don't need that kind of negativity in your life.