I think the key to this is properly defining marriage in the eyes of Jesus. As stated in the below text, Jesus defines marriage as having intercourse with one another:
What? know ye not that he which is joined to an harlot is one body? for two, saith he, shall be one flesh. But he that is joined unto the Lord is one spirit. (1 Corinthians 6:16, 17 KJV)
We can see here the correlation of man joining himself to a woman just as Christ joins himself to us. Now clearly this is speaking of marriage for a number of reasons, but not least of which is the fact that we are the bride of Christ and he is relating how the man joins himself to the woman.
However, let's concrete this a little more by going back into the old testement and prove that intercourse is in fact the marriage of a man and woman.
And Judah said unto Onan, Go in unto thy brother's wife, and marry her, and raise up seed to thy brother. (Genesis 38:8 KJV)
You see the world today gets ceremonies mixed up with everything. Marriage has nothing to do with a ceremony, and nothing to do with mans laws, in the eyes of Jesus.
And the Pharisees came to him, and asked him, Is it lawful for a man to put away his wife? tempting him. And he answered and said unto them, What did Moses command you? And they said, Moses suffered to write a bill of divorcement, and to put her away. And Jesus answered and said unto them, For the hardness of your heart he wrote you this precept. But from the beginning of the creation God made them male and female. For this cause shall a man leave his father and mother, and cleave to his wife; And they twain shall be one flesh: so then they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder. (Mark 10:2-9 KJV)
Now, with this scripture we can finish the foundation of this answer. Jesus is speaking to the Pharisees and explaining marriage a bit more to make it clearer. They saw marriage just like the world sees it today, legalistically. However, as Jesus stated, from the beginning it was defined as a man and a woman cleaving to each other and becoming one flesh (which finishes lining up the scripture from Corinthians).
However, because of the hardness of mans heart Jesus was forced to allow a writ of divorcement so that the institution of marriage (as defined by two becoming one flesh during intercourse) could continue. You see, the Bible stated from the beginning the following:
And the Lord God said, It is not good that the man should be alone; I will make him an help meet for him. (Genesis 2:18 KJV)
And that hasn't changed just because two people are incapable of staying together and resolving differences for whatever reason. Therefore divorce had to be instituted so that marriage could continue. But the definition of marriage, as you can see from when Jesus defined it, has not changed.
Jesus Christ the same yesterday, and to day, and for ever. (Hebrews 13:8 KJV)
So, now that we've got that out of the way we can answer the question, what do those that involve Jesus in their marriage have that others don't? Simple, they have a foundation and a rock to hold them together through the hard times and help them make good decisions as He guides them through their lives. But marriage itself has nothing to do with a priest, a law, or anything else - just two coming together as one.