Why did the Dread Pirate Roberts kill Vizzini?

The reason for this is because the Dread Pirate Roberts was actually a good man and a man of honor.

When meeting Inigo he is promised that he will not be killed before reaching the top, with Inigo even swearing on the soul of his father and throwing a rope down to help Roberts to climb. Once on the top he again urges Roberts to wait until he is ready to fight him. These are clear indicators that Inigo is a man of honor and not a man of evil.

After that Inigo tells him the story about his search for vengeance, showing that he is only on his current path for a rather noble goal instead of something like monetary gain.

When he meets Fezzik he could kill Roberts in a single hit, but decides not to because he wants to fight a fair fight, man against man no weapons. Even during their initially one sides fight Fezzik shows compassion and friendship instead of malice. Again showing that he is a good guy, just on a wrong path like Inigo.

Vizzini isn't on the wrong path while being a good guy because he chooses the path, he chooses to be the bad guy. And of course he cheated... as was expected... So why show mercy? If shown Mercy Vizzini would probably haunt him to get revenge while the other two were beaten honorably and had no motivation to pursue Roberts now that Vizzini isn't there to order them.

And incase somebody would say that Roberts also cheated, Roberts only asked him to deduce where the powder was. He never stated that only one of the goblets would contain it, the fact that the rules were not made clear is how he won the battle of wits.

A.bakker has a great answer, but misses an important point. The Dread Pirate Roberts is a pirate, and one with a very specific reputation.

Fezzik and Inigo are obvious underlings and just following orders, yet they do show honor.

Vizzini, on the other hand, takes pride in being a snake. He has no problems showing his bad side (as that might be the only side he has, but we don't know enough about him to really know this). Also, when Roberts meets Vizzini, he has a knife to Buttercup. He also treats her roughly even after they agree to a match of wits, which also shows that he has no honor and that he really only cares about himself. This is also evident by Vizzini leaving his men to try to slow down Roberts. I also seem to remember some snide remark about the other two losing, when Roberts arrives, showing his lack of empathy. (It's been too long since I watched the movie or read the book.)

Then there's the fact that Vizzini has the kind of general braggadocios attitude that makes even the most gentle of people want to punch him. With all of this evidence, it doesn't take much for someone who is known as a ruthless pirate to think, "The world would be better without this guy. And he's bumping up against my ego, anyway."

And what better way to show a guy like Vizzini up than by setting him up for something he simply can't win. As for making it lethal, there's nothing to say an SOB like Vizzini won't go back on his word and still go after Buttercup again. Killing Vizzini just guarantees no more problems.

Edit

After watching the movie again last night, Vizzini is the one that suggests it's lethal, as others have mentioned. This is after he brags that Roberts can't match his intellect, so of course Roberts thinks differently and suggests a match of wits, trying to get Vizzini's ego involved in his own downfall.

Vizzini posed an immediate danger to Buttercup...

Fezzik and Inigo posed no immediate threat to Buttercup, and their behavior suggested to Westley that they probably would not kill her even if they had the chance. They both showed themselves to be people of honor who would shun killing in an unfair circumstance. Vizzini on the other hand had a knife to Buttercup's throat and had shown no inclination to spare the helpless. Westley could not mess around in this situation. Why did he not use an incapacitating drug instead of poison? Perhaps he didn't have any. Neutralizing Vizzini was an absolute strategic necessity in this situation. It was not just life and death, it was Buttercup's life and death.

...and Westley also needed to keep his honor intact.

Westley challenged Vizzini to the battle of wits and accepted Vizzini’s proposal that it be to the death. These terms were the only way to get Vizzini to lower the knife, and once the challenge was accepted, killing Vizzini in a battle of wits was the only way to maintain his honor. After all, it would have been easy for Westley to lunge across the stone and overpower Vizzini the second he put the knife down, but that would have been dishonorable. The way in which Inigo, Fezzik, and Westley in the final battle with Vizzini eschew quick and easy solutions to their challenge in favor of honor and fairness is what marks them as heroes. Further, honor could play a role in Westley not using a drug instead of poison--the challenge after all was "to the death." This strict adherence to honor and being as good as his word makes it carry even more weight later in the movie when he challenges Humperdink to a fight "to the pain" (even if it was a bluff).

Looked at from a bird's eye view: Vizzini's death falls into the "just deserts" category. He's a difficult sort who could probably make Gandhi want to punch him in the nose (so very little emotional connection with the audience; they won't be disappointed to see him go). He shows no concern for putting others at risk to further his own goals; he's not just reckless, he gleefully uses their danger to his advantage (again, the audience will be ok with his death - I certainly laughed).

A delightful summary of how this works in plays and movies is in "Rosencrantz and Guildenstern Are Dead". Guildernstern talks with an actor who unpacks drama's guiding tricks and principles (from within a play inside the play - cool trick that is pulled off without a hitch):

Player: Events must play themselves out to aesthetic, moral and logical conclusion.

Guildenstern: And what's that, in this case?

Player: It never varies — we aim at the point where everyone who is marked for death dies.

Guildenstern: Marked?

Player: Between "just desserts" and "tragic irony" we are given quite a large scope for our particular talent.