There’s a lot of controversy over these two concepts in the Christian world. Does submission require obedience? Or is it possible to submit without obeying? Consider this event recorded in Acts:
27 Having brought the apostles, they made them appear before the Sanhedrin to be questioned by the high priest. 28 "We gave you strict orders not to teach in this name," he said. "Yet you have filled Jerusalem with your teaching and are determined to make us guilty of this man's blood."
29 Peter and the other apostles replied: "We must obey God rather than men! 30 The God of our fathers raised Jesus from the dead — whom you had killed by hanging him on a tree. 31 God exalted him to his own right hand as Prince and Savior that he might give repentance and forgiveness of sins to Israel. 32 We are witnesses of these things, and so is the Holy Spirit, whom God has given to those who obey him."
33 When they heard this, they were furious and wanted to put them to death. 34 But a Pharisee named Gamaliel, a teacher of the law, who was honored by all the people, stood up in the Sanhedrin and ordered that the men be put outside for a little while. 35 Then he addressed them: "Men of Israel, consider carefully what you intend to do to these men. 36 Some time ago Theudas appeared, claiming to be somebody, and about four hundred men rallied to him. He was killed, all his followers were dispersed, and it all came to nothing. 37 After him, Judas the Galilean appeared in the days of the census and led a band of people in revolt. He too was killed, and all his followers were scattered.
38 Therefore, in the present case I advise you: Leave these men alone! Let them go! For if their purpose or activity is of human origin, it will fail. 39 But if it is from God, you will not be able to stop these men; you will only find yourselves fighting against God."
40 His speech persuaded them. They called the apostles in and had them flogged. Then they ordered them not to speak in the name of Jesus, and let them go.
41 The apostles left the Sanhedrin, rejoicing because they had been counted worthy of suffering disgrace for the Name. 42 Day after day, in the temple courts and from house to house, they never stopped teaching and proclaiming the good news that Jesus is the Christ. (NIV)
Here we see both submission and yet disobedience to the Sanhedrin. They submitted by not resisting the punishment for their disobedience. It is possible to both submit to authorities and yet disobey them. The disobedience should be for the right reason, however. Consider this:
1 Peter 2:20-21
20 But how is it to your credit if you receive a beating for doing wrong and endure it? But if you suffer for doing good and you endure it, this is commendable before God. 21 To this you were called, because Christ suffered for you, leaving you an example, that you should follow in his steps. (NIV)
This was the way of the disciples. A modern day example was Martin Luther King, Jr.