Explanation: Encantado means you are delighted to meet them. Mucho gusto means nice to meet you. El nombre would be refering to saying anothers name most likely a male. But, me llamo is refering to your own name.
'Mucho gusto' is what is commonly spoken -- though 'encantado' is still technically correct; it's quite old-fashioned or outdated. (And when it <em>was</em> common, say 50 - 100 years ago, a man would say it to a woman, not a man to another man.)
