the difference between baptism of john and that of paul
This is somewhat outside the area where I have studied. So you will probably want to ask someone else as well.
John's baptism was a baptism of repentance. Thus, a person had to repent of his sins to be eligible for baptism. Persons who had been baptized with the baptism of John were re-baptized by the Apostles, on request. From my recollection, so were people who had been baptized in the name of Jesus only.
I think one reason people ask about Paul's baptism is the question of whether or not to baptize infants. This is one thing I have studied.
Paul's baptism followed Jesus' command to baptize all nations in the name of the Father, Son, and Holy Spirit. Paul baptized an entire household. You may be aware that Lutherans argue that most households at that time contained children, and that Paul baptized the children as well. This is not definitive; however, there is nothing in the passage to suggest that baptism was withheld from anyone because of age (being too young). There is nothing in the Bible anywhere that says that baptism should be denied to children of any age. Most people will argue that very young children cannot understand, but they are forgetting that understanding is SPIRITUAL, and thus, since the Bible indicates people receive the Holy Spirit at baptism (if they haven't already), the infant will now have the Holy Spirit to give him understanding.
Peter writes that baptism saves us, in like manner as the eight souls were saved in the ark, traveling through water. The saving by traveling through water in the ark is a symbol of baptism. This is another passage on which Lutherans rely when advocating infant baptism. To determine my view of this passage, I looked at the original Greek. The verse also talks about a good conscience calling on the name of the Lord, but there is no definitive indication that this comes BEFORE the baptism.
The Bible appears to be silent on the details of John's baptism. Likewise, there is no significant discussion of why people needed to be re-baptized if they had received only John's baptism. The Bible does not prohibit children from being baptized. The command is to "all nations", not "all nations except children". Further, Jesus commanded people to let little children come to Him and forbid them not. Since Jesus is no longer physically present on earth, there is really only one way little children can be brought to Jesus, and that is through baptism. Dedication, which anabaptists practice (i.e. those who reject infant baptism, such as Baptists and some evangelicals), is an Old Testament practice, and there is no indication it was carried forward into the church.
So the question must be asked, do we have any RIGHT to FORBID little children to be baptized? I would say we do not.
In the Old Testament, boy children were circumcised. Nobody asked them, and nobody expected them to understand or consent. It was a requirement. Peter thought that Christian men should be circumcised, but the Apostles, under the guidance of the Holy Spirit, rejected this along with any other attempts to Judaize people, and require them to keep the 613 commandments of Moses. Only 9 of the 10 commandments were re-stated by Jesus (the sabbath commandment was omitted). The early Christians worshiped on Sunday to commemorate Jesus' resurrection. So we are only enjoined to observe the 9. We are told not to forsake gathering together, but as Paul states in Galatians, the day is up to us. In addition to circumcision, people who had become ritually unclean were required to dip in the waters of the mikva. There was no exception for very young children who had become ritually unclean. The Old Testament lists a number of different things that can make a person ritually unclean. The washing in the mikva foreshadows Christian baptism.
I also personally hold that a non-Christian, including an adult, who is struggling with believing, is a candidate for baptism WITHOUT REQUIRING A CONFESSION OF FAITH. I have seen in action that baptism can enable such a person to believe. I had a rather startling and dramatic experience of this in my own life, when someone close to me was baptized. While the Lutheran church doesn't teach this, and requires instruction (which is entirely proper as far as I am concerned), I would part company with the idea that a person who is struggling is anything but a spiritual infant, and thus that he is not eligible for baptism on request. It took years to find someone willing to baptize a person who was struggling, but what a difference it made when it was finally accomplished! Matthew 28:19 does NOT SPECIFY an ORDER in which teaching an baptism are to occur. Both are to be done. I believe the order in which they occur should be the choice of the believer who is teaching and baptizing, guided by maturity in the faith arising out of the indwelling of the Holy Spirit.
So I would hold that while John's baptism was a foreshadowing of the baptism to come, Paul's baptism is available to all people regardless of age, and it is in the name of the Triune God.
If this doesn't answer your question, please do feel free to ask another of the experts.