What were the main differences are between the historians Tactius and Livy? 

The most important difference between these two writers is that Tacitus was far more critical of the Romans than was Livy. Livy's intent in his writings was to, in his words, "do my duty to the memory of the deeds of the most important people on earth." The "most important people on earth," of course, were the Romans and their leaders, and his accounts are mostly hagiographical, celebrating the lives of great Roman republican statesmen and leaders. Tacitus, on the other hand, is more ambivalent about the exercise of Roman power. In particular, he was quite sympathetic to many of the Romans' enemies, and critical of tyranny. The Germans, for instance, are praised as an "incorruptible," virtuous, and free people, contrasted with the corrupt Romans.