Why Christians wear some elaborate costumes which sometimes outshine those of the pre-Christian epoch? What is the need to use elaborate costumes at services and on other occasions? Costumes of some Christian confessions much outshine those of some other religions, including the pre-Christian ones. 

Some of the elements can hardly be traced to pre-Christian times and seem to be Christian inventions, while others can be clearly traced to polytheistic religions of Romans and Greeks or even ancient Egypt, like the Crosier:



Yet it seems that pre-Christian dress of the Roman and Greek priests were much simpler, functional and ascetic.

In Orthodox Christianity they use for service a headware called mitre . 

The corresponding prayer when putting it on mentions "expensive jewels" which is probably a requirement for this kind of vestment. If it is used to show respect to God as some answerers put it, my question is whether wearing expensive Rolex watches will also show one's respect to God? 

So what the purpose of those elaborate costumes, especially in Catholic and Orthodox church?
In short: 


Dressing nicely in front of God is a good thing.
Priests do not actually dress in an "elaborate" way, but they do dress in a formal way.
Some people see pictures of the pope during very special ceremonies, and assume that all priests dress this way, which is simply not true. The pope is required to dress in a way that fits his status as pope. He is the Vicar of Christ, and Catholics expect him to dress like it. This does not mean that his garments must always be ornate, but that they should be appropriate. But this question is about priests, not the pope.


So why should priests wear "special" or "nice looking" clothing? Some of the basis for this is:


  John 12
  3 Mary took a liter of costly perfumed oil made from genuine aromatic nard and anointed the feet of Jesus and dried them with her hair; the house was filled with the fragrance of the oil.
  4 Then Judas the Iscariot, one [of] his disciples, and the one who would betray him, said,
  5 “Why was this oil not sold for three hundred days’ wages and given to the poor?”
  6 He said this not because he cared about the poor but because he was a thief and held the money bag and used to steal the contributions.


We should not hesitate to make our priests match their position. They are serving God. We dress nicely when we go to meet someone important to us, and priests should do the same. Sometimes, the Catholic priest acts in Persona Christi, which means "in the person of Christ". The priest does not become Christ, and he does not become a high priest, but he exercises certain very sacred powers that have been granted to him by God. We should have respect for that, and we should think of the eternal heavenly service of the High Priest Jesus during Mass, especially during the Eucharistic Prayer. From the Baltimore Catechism:


  
  365. Who said the First Mass?
  Our Divine Saviour said the first Mass, at the Last Supper, the night before He died.


The vestments themselves are not actually "ornate", even when the are on the more-elaborate side of the scale. They are simply formal. For example:



The dress goes a long way towards communicating the importance of religious life, and the commitment of the priest to living that life. Appropriate priestly vestments lend dignity to the office of the priest, and they lend dignity to the services themselves:



Catholics generally like to see that sort of formality. It is beautiful, and beauty praises God. Not your taste? That's fine, there are smaller Catholic parishes. But there is nothing wrong with giving praise to God in this way. Jesus might wear humble clothes, but that does not mean that we should wear our most "humble" clothes when going to see Him. He is the King of Kings, the Lord of all creation.