Let us first briefly review the sequence of events that occurred just before Jesus was brought before Pilate. While he and the disciples are in the Garden of Gethsemane Judas arrives with armed officers and some religious leaders. As a sign of whom to arrest Judas kisses Jesus (Mark 14:41 - 45, John 18:1 - 8).
Jesus is arrested and taken to the court of Annas, a former High Priest and father-in-law of the current High Priest Caiaphas. After some questioning, he is sent to the High Priest's palace. Gathered at the palace are those, and only those, chief priests, elders, scribes, and other religious leaders who want to see him dead.
After a farce of a trial is held, complete with false witnesses, Caiaphas accuses Christ of blasphemy. Caiaphas and all those gathered agree that he deserves death (Matthew 26:59 - 68, Mark 14:55 - 65). Another rigged trial, this one at 5 am, is held before some religious leaders. He is then taken to Pontius Pilate around 6 am.
The original charge of blasphemy against Christ is changed by the Jewish leaders to treason against Rome when they take him to the Roman authorities (John 18:33 - 36, HCSB). Christ, after questioning, is then sent to Herod (Luke 23:7). The response he gave to the many questions he was asked was silence (Luke 23:8 - 9).
It is true that Scripture does not explicitly say why Jesus answered one authority but not another. It is also true that he, during his two trials before the Jewish leadership and then questioning by Pilate, generally avoided saying anything except when He was ordered to (see Mark 14:60 - 62; Matthew 26:62 - 65). He behaved this way in order to fulfill the prophecy in Isaiah (Isaiah 53:7).
Not life or death trial
It should be noted that when Jesus stood before Herod Antipas he was not on trial for His life. Antipas not only thought he didn't have the final authority over his fate he wasn't particularly interested in executing Him either. This is why he sent him back. Pilate, as Roman Prefect of Judea (26 - 36 A.D.), had the ultimate local civil authority over all criminal court cases in his jurisdiction (see John 18:31). He could not ultimately dodge his responsibility, although he certainly tried hard to have others do his job for him!
Although Jesus had spoken earlier to Pilate, He was silent during his second appearance before him (John 19:8 - 11). The Prefect had to threaten him to get a response, since he was totally determined to make sure He would lose his case. Christ responded to him because he had ultimate legal power over Him.
The classic Bible commentator Matthew Henry makes an interesting point about why Antipas likely did not get any cooperation from Jesus. Herod wanted to see him perform a miracle in front of him, "on command" as it were. Our Savior, however, did not do miracles as a means of entertaining or amusing others like a modern magician doing magic tricks.
Herod, in the few years before this meeting, could have gone out into the countryside to meet the Lord during His ministry just like any other person. He not only would have likely seen a miracle but also would have been able to inquire directly about his teachings had he been interested.
On the day of his crucifixion, however, Herod joined his soldiers in mocking Christ. He then sent Him back to Pilate to handle. By doing this, he showed respect toward the Roman authority. Admittedly, although it is not fully clear why Christ did not answer Herod, the explanations given above are likely involved in why this occurred.