Why do monastics in certain Buddhist traditions take new names? In the Theravada tradition, and perhaps others, monastics take a new name when becoming bhikkhus/bhikkhunis. Where does this tradition come from, and what is its significance? Others don't - what's the difference?
In India, during the time of the Buddha, the vast majority of monks and nuns didn't change their names. Notable exceptions are Sariputta and Moggallana, whose original names were "Upatissa" and "Kolita" respectively.

It is not clear why the Buddha chose to call Upatissa by the fact that he was the son (putta) of Sari, or Kolita by the name of his village. The Mahavagga calls them by these names before they met the Buddha, so it may have been that they took these names earlier. 

Other examples that may also have been acquired earlier are Kundala-Kesi, Bakkula, etc.

Cases of adding a descriptive to the name are common, e.g. Kumara Kassapa, Lakuṇṭaka-bhaddiya, etc.

One case stands out, that of Angulimala, who actually had his name changed by the king:


  Then the king's fear, his terror, his hair-standing-on-end subsided. He went over to Ven. Angulimala and said, "Are you really Angulimala, lord?"
  
  "Yes, great king."
  
  "What is your father's clan? What is your mother's clan?"
  
  "My father is a Gagga, great king, and my mother a Mantani."
  
  "Then may Master Gagga Mantaniputta delight [in staying here]. I will be responsible for your robes, almsfood, lodgings, &amp; medicinal requisites for curing illness."
  
  http://www.accesstoinsight.org/tipitaka/mn/mn.086.than.html


Apart from all this, the biggest reason for later name changes is simply because names in other languages don't work for the Pali recitation required for ordination, e.g.:


  ayaṃ Noaho āyasmato Tongassa upasampadāpekkho


doesn't really work, so giving monks Pali names is considered appropriate:


  ayaṃ yuttadhammo āyasmato sirimaṅgalassa upasampadāpekkho


It seems that early on there was a tradition of naming monks after arahant disciples of the Buddha, so we get various Moggallanas and Sariputtas throughout the ages. In modern times, there seems to be some greater variation in the practice, and choosing a name for a monk has taken on significance that pretty clearly wasn't there in the time of the Buddha. 

My name, for example, is chosen because I was born on a Wednesday - in fact, I was given the wrong name, because one monk thought that me being born in the morning in Canada meant that he should give me a Wednesday night name (it having been night in Thailand when I was born). 

In the end,


  What's in a name? That which we call a rose
  By any other name would smell as sweet;
  So Romeo would, were he not Romeo call'd,
  Retain that dear perfection which he owes
  Without that title.
  
  -- William Shakespeare, Romeo and Juliet