Just poking around on YouTube, I found this compilation of scenes from the second Star Trek pilot, "Where No Man Has Gone Before," which I find -- if you'll pardon the expression -- quite fascinating.
I apologize for the poor picture quality -- I'm not sure what's going on.
The only really new-to-me scene is the first one, where Kirk makes this log entry:
Enterprise log, Captain James Kirk commanding. We are leaving that vast cloud of stars and planets we call our galaxy. Behind us, Earth, Mars, Venus, even our sun are specks of dust. Question: What is out there in the black void? Until now our mission has been that space law regulation, contact Earth colonies and investigation of alien life but now, a new task, a probe out into where no man has gone before.
Other than that, the only differences are the Act announcements at the start of each scene, and the radically different theme music.
You can also see fairly easily the zippers that run up Kirk and Spock's left shoulders to the collars. Apparently all the shirts zipped up like this, but Nimoy's in particular had to be larger so they could protect his makeup: I think they could only afford so many rubber ears per season.
See if you can spot George Takei in the scene under the opening credits. Hint: he's wearing a different shirt than usual, since he's credited as "Chief Physicist" in the closing credits. Second hint: If you hit the elevator scene, you've gone too far.
Speaking of the credits, I note that the only character who's named in them is Spock. Of the rest, only George Takei (Chief Physicist) and James Doohan (Engineering Chief) are names I recognize.
Also, it's dated MCMLXV -- 1965. I think these scenes must be from whatever version they screened to NBC after the network requested a second pilot.
EDIT: For comparison, you can watch the broadcast version here. There are advertisements, unfortunately, and often they cut in on the middle of one of Shatner's gripping deliveries.