Observation: This discussion is well-understood as an attempt to trap Jesus with a no-win question. If he responds saying that Jews should not pay the tax, the Pharisees can report him to the Roman authorities for sedition. If he says Jews should pay the tax, the Jews will be furious. Jesus will give a brilliant response. However, one of the issues of this question that is often uncommented upon is the ways the Pharisees characterize Jesus himself, in the hopes of forcing him into a specific context. Specifically, let's look at four statements they make about Jesus.
First, the call him a man of integrity. This puts him in his place. It denies him 'title' as rabi or teacher or prophet, and effectively says he's just a good man trying to do and say what is right.
Second, they say he is not swayed by others. This is provably untrue. Several times during his ministry, Jesus has responded to requests explicitly because he was swayed to heal or help or advise. He has even declared he will not do something, then changed his mind when another person gives an explanation for their request.
Third, they say he pays no attention to who others are. Again, this is false even in the immediate context, and certainly in the heavenly context. As God incarnate, Jesus absolutely pays attention to everyone. And if this statement is about not being impressed by others' titles or positions, right here and now - in this very setting - Jesus acknowledges who are Pharisees and who are Sadducees. He pays attention and responds according to who they are.
Fourth, they say Jesus teaches the way of God in accordance with the truth. This is a misleading characterization because of the Pharisees' false understanding of the truth. Jesus does teach the way of God, but he does so in accordance with the word of God. Jesus reveals the ways of the Lord exactly as they are revealed in scripture, and rejects the layers of false truth the Pharisees teach.
Ultimately, Jesus' response - to give back to Caesar what is Caesar's and give to God what is God's - not only turns the table on the core question, but it refutes all four of these characterizations. His answer shows him to be more than a man, but truly a wise teacher; It shows him to be swayed by others, as he is keenly aware of, and calls out, the nature of the trap before him; It shows him to be paying attention to others, as he embeds his answer within the context of the Emperor; It shows him to teach with the context of scripture and with the Lord central in your opinions and thoughts, instead of being concerned about rules and "truth" created by human authorities.
Application: This is a deeper understanding of Jesus' response that I never considered. Yes, it is a common technique in debate to create a false context or set of 'givens' and then trap your opponent into responding within that context. However, I have never before seen that Jesus didn't just avoid the trap ... he overturned the false context. No wonder the Pharisees were "amazed" by Jesus' response.
People forget that Jesus was and is so smart and wise that he can prove himself to be intellectually superior to an entire group of scholars, without any effort. Of course, as God, he knows everything. It is just a reminder that he truly is that great and that wise and that amazing.
Prayer: Jesus, you are more than my Lord and savior, you are my wise and brilliant teacher. I know I am often slow to learn, if not downright stupid. This day, may I rely on your wisdom and intelligence, and not on my own thoughts, words, or understanding. Amen.