From this point to the end, Scorsese followed the script pretty faithfully. He goes to the firing range, burns the flowers, and tells his diary, "My whole life has pointed in one direction." He sends money to Iris and tells her, "By the time you get this, I will be dead." Scorsese switched the order of these two scenes, but otherwise they play faithfully.
He goes to the third rally and sticks his hand into his jacket as he approaches the platform. The Secret Service agents rush him and he escapes. The only difference is that Scorsese gave him a Mohawk haircut. Schrader described him as "…the most suspicious human being alive. His hair is cropped short, he wears mirror-reflecting glasses. His face is pallid and drained of color, his lips are pursed and drawn tight…he looks sick and frail."
Schrader had Travis go home and check the mail to find that the letter to Iris has gone before he strips to the waist and walks back and forth in his apartment.
There's another difference here. In the movie Sport pays the customer who goes up to see Iris, so I figured he was Sport's connection. In the script, he's a private cop and there's no mention of money being exchanged. I think Scorsese felt it would be too confusing if there wasn't something clearly off about this customer, so he showed the pimp paying the john.
Travis shows up and shoots Sport. Scorsese didn't use most of the dialogue that Schrader wrote, but did include, "suck on that." He also had Travis sit down on the steps of the building for a while, maybe waiting to see if anything happens, or maybe in disbelief that he just shot a man. In Schrader's world, he goes straight into the building.
Schrader had Travis blow off the old man's whole hand, but I thought just blowing away most of the fingers was more shocking.
Sport shoots Travis, Travis shoots Sport, the private cop shoots Travis, and the old man comes after Travis. This is all the same.
"Travis, trapped under the heavy Old Man, reaches down with his right hand and pulls the combat knife from his right calf. / Just as Travis draws back the knife, the Old Man brings his huge left palm crashing down on Travis: the Old Man's palm is impaled on the knife." Same.
Iris is screaming, police sirens are screaming, and Travis picks up the last gun: the one carried by the private cop. There's dialogue here. The old man begs for his life and Iris screams, "Don't kill him, Travis! Don't kill him!" I honestly don't remember any dialogue: just horrified screaming. But there might have been dialogue.
The cops arrive. "He forms his bloody hand into a pistol, raises it to his forehead and, his voice croaking in pain, makes the sound of a pistol discharging." The only thing Scorsese added to this was the blood dripping from his finger.
Schrader even called out for an "OVERHEAD SLOW MOTION TRACKING SHOT" here and Scorsese used it.
Fade to all the stuff in Travis' apartment about the brave cabbie battling the gangsters to return innocent little Iris to her parents and the letter from Burt Steensma.
And finally we see that Travis has returned to work and grown his hair out. Betsy gets into his cab and they discuss how brave he was. Then Schrader writes, "CAMERA FOLLOWS TRAVIS' taxi as it slowly disappears down 56th street," and that's the end. Scorsese put in one more shot of Travis' eyes in the rear view mirror and a tiny hint that all was still not quite right in his world, which again made the Travis we met more sinister than the one Schrader wrote.
Schrader wrote Taxi Driver during a very low point in his life and Travis was drawn from all the bad things he felt within himself at the time. I think this is why he tells us a lot that Travis is drifting towards violence, but pulls back from making him as sinister and violent as he could in dialogue and action. He could not quite follow through with making this character as insane and out of control as Scorsese and DeNiro did because that might mean that he was that insane and out of control. Or maybe he did feel that insane and out of control and was unable to fully explore it out of fear.
But Scorsese and DeNiro had already made Mean Streets together. They have always been fearless about bringing it all out on the screen.
Much of Schrader's dialogue skates over the issues, or doesn't explore them at all. The dialogue in the first scene with Betsy in the diner was completely re-written for the movie. Schrader was probably the one who re-wrote it, but he did it as Scorsese's direction, and very likely worked with DeNiro and Shepherd while he did, so it had way more impact in a much shorter time than what he'd written originally.
Skating over the issues worked during the scenes between Betsy and Tom, and Scorsese left them alone, but he did a lot of tinkering with the dialogue between Betsy and Travis.
He also left most of the diary entries alone. Schrader couldn't bring himself to make Travis as sinister with other people as Scorsese did, but he did manage to find that very dark place when he and Travis were alone together.