#Why didn't the computer realize Picard was elsewhere

 - If com badges are used to track people, they're clearly not a
   reliable identification system and it would be a bad idea for the
   computer to stop accepting commands from Picard simply because his
   badge were stolen or damaged).
 - Data is not the average attacker. He's able to manipulate the Enterprise's computers almost at will, circumventing security mechanisms and setting up ones of his own. It's thus possible that he simply disabled any such sanity checks, even if they existed (i.e. if the computer had used biometric-based location tracking, e.g. heart rhythm/gait patterns to identify crew locations).

#Why TOS-era computers could identify faked voices but not the Enterprise-D

 - Security is an arms race, often biased towards attackers rather than defenders. At various times, one side will outpace the other, and it's possible that TOS era biometric security was stronger than voice reproduction technology (or at least in that specific instance against the Eminar VII voice duplicator).
 - Data is not your average robot. Doctor Noonian Soong was quite adept and very meticulous when it comes to replicating biological/humanoid characteristics in his androids, to the point that [at least one of his androids][1] has lived for decades amongst humans undetected (and without her even knowing it herself). Data is built with hair that grows, simulated breathing, a life-like blinking algorithm, an aging program, a circulatory system, the ability to dream and get drunk, and other human-like anatomic details. It's therefore plausible that his speech synthesizer has the ability to fool most biometric systems.

 Also remember, Data's an exceptionally intelligent and resourceful person, so using his wide base of scientific knowledge, his superhuman analytic abilities and attention to detail, and auditory processors and sensors that likely rival the ship's own voice analyzer, he would probably be able to precisely analyze Picard's voice to the same level of detail as the computer and deliver a reproduction that was just authentic enough to pass muster.

  [1]: http://en.memory-alpha.org/wiki/Juliana_Tainer