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Probably because this would seriously break Vincent's routine and draw unwanted attention.
As Vincent points out, Max must visit his mother in the hospital, because he does every day. Max not showing up would be noticed, elicit suspicion, and, in the worst case, would cause people to look for Max, which could seriously hinder Vincent's operation. The same applies for the regular calls Max gets on his radio from the cab central, which he has to answer himself.
Part of Vincent's efficient modus operandi is that he keeps as invisible as possible while he is doing his work, and the best way to accomplish that is by remaining anonymous and not drawing attention. The case that Dt. Fanning recalls about the other cab driver who supposedly killed several people and then killed himself, strongly suggests that it was Vincent on an earlier assignment. If it was Vincent, he kept a successfully low profile. Killing Max before Vincent completes his tour could therefore seriously disrupt Vincent's schedule, so he has to take care that Max keeps doing his normal routine, and not kill him before the end. Edit (Coming Soon)
When Max sees Vincent kill the two men that stole his wallet and Vincent's briefcase after Vincent handcuffs Max to his steering wheel to incapacitate him while he performs his second hit of the night, Max lowers his head in despair and says "fuck!". On first glance it appears that it upsets Max to see a man murder two others with such ruthless efficiency and disregard in front of him, even if they are the ones who stole his wallet at gunpoint. However, a second reason might be that Max was hoping that the muggers would kill Vincent so Max could be free, but when Vincent turns the tables on the two muggers and kills them with ease, Max knows that he is in it for the long-haul. Though the most likely explanation is that Max, an average-Joe cab driver just saw two people murdered right in front of him, also because Max is the one that drew them to the car, he may have felt partially responsible for their deaths. Remember that Vincent says when they are getting gas "If you draw attention to yourself, you're going to get people killed who don't need to be." Edit (Coming Soon)
A film making reason could be that it would have removed about 25 minutes from the second act of the film, had Max destroyed the list and Vincent didn't care because he knew the rest of the targets. However, a movie explanation could be that; seeing as how we don't know how detailed the files were on each target, we have no idea how much information was available to Vincent for each target. There could have been far too many details to memorize in such a short time (keep in mind, Vincent only got the information on his targets when he was leaving the airport). There could have been information such as where each target would be at a particular time of day, the best way to get access to them such as identifying himself as a notary to get access to Clark, knowing that Daniel would love to sit and have a drink and boast about jazz and meeting Miles Davis after his club was closed, that Peter Lim would be at a night club and so on. It's also possible that the information was updated every so often by people staking out each target. So when Vincent checked his laptop for the next target, he had current, up-to-date information. Edit (Coming Soon)
While the movie leaves this ever-so slightly ambiguous, the answer is no.
The reasons are; during the club shoot-out, there were dozens of witnesses that would have seen a grey-haired caucasian man doing the majority of the shooting and would not indicate that Max had anything to do with them. Fanning had voiced his thoughts to the FBI and to his partner about the theory that someone took a cab driver hostage and killed people, then murdered the cabby to make it look like a murder-suicide, so this is something they would keep in mind. There would be sufficient fingerprints, hair fibres, DNA at Max's crashed cab that belonged to Vincent, there was a shoot-out at Annie's office which once again would leave DNA evidence by Vincent and then show that they were chased by Vincent. Vincent killed a security guard and took his gun, evidence would prove that he had the gun and was using it at the time of his death. There were witnesses watching Vincent stalk Max and Annie on the train, the FBI thought that Max was "Vincent" and had actually killed the "real" Max because he looked like the cab driver, but minimal investigating would prove that Max was indeed himself. There would be Annie's testimony that Max rescued her from a hitman, the same one who was laying dead on the train and whose body could be linked to all the previously mentioned DNA evidence. And finally Max's testimony that he was taken hostage and forced to cooperate with Vincent.
The only real things that suggest that Max may have been convicted of the killings is that the FBI were convinced he was in fact the killer, Fanning; the one person who believed Max was shot dead outside the club, the FBI had Max on video saying his name was Vincent and have him going in to meet Felix. When Max crashes the cab, a cop sees a dead body in the trunk, Max disarms him and pulls the gun that Vincent had used to do all the killings with (with the exception of Daniel) and Max continued to use this gun for the remainder of the film.
Bottom line: the evidence proving Max was innocent far outweighs that which would prove him guilty. Edit (Coming Soon)
If the story that Fanning tells of the cab driver who went crazy and shot a bunch of people and then himself was actually Vincent on a previous mission, the most likely answer would be yes.
The contradicting evidence is that Vincent says numerous times about what Max should do 'when' the night is over. But Vincent is a trained killer, he would know that giving Max hope by talking in such a way that would suggest that Vincent would let him live, would keep Max from doing anything stupid. Vincent has shown that he is perfectly capable of ruthlessly killing a person with whom he just had a drink and a pleasant conversation, so he may not even feel disconcerted in the slightest way by the idea of killing Max.
However, the banter between the two of them could have shown Vincent that Max was a human being, and Vincent felt a connection to him and didn't want to kill him. At the very least it's likely that's what caused Vincent to hesitate when they were facing off on the train and allow Max to get the first shot off. Not to mention the fact that had Vincent truly wanted to kill Max at the end, he still had time to pick up his extra mag, reload and shoot Max. But instead he accepted that Max had beat him, sat down and had one last friendly chat with Max until he died. Edit (Coming Soon)
The obvious reason would be that Vincent didn't know that Felix ordered hitmen to kill him if anything went wrong at Club Fever. Vincent didn't know the difference between Felix's hitmen and Lim's guards. He assumed that they were all Lim's guards, and therefore they must all be eliminated for him to kill Lim and move on to the last hit of the night.
Also, Vincent may have assumed the Mexican hitmen weren't Lim's guards (as Lim and all his crew were Asian) which is why Vincent sent Max in to pose as him in the first place. Vincent knew that Felix probably wouldn't be too happy that he lost the list and may want to kill Vincent. So had Felix's men killed Max in the club, Vincent wouldn't have the list so he'd likely just leave town. But if Max succeeded in getting the list (which he did) and Max was killed by Felix's hitmen, Vincent still had the list and would complete the job (as his reputation was on the line). Felix would realize he killed the wrong man, but the real Vincent followed through on his assignment, thus keeping his reputation and his anonymity intact. Edit (Coming Soon)