Both words look and sound very similar but they are the exact opposite. Nritta is pure technique. It incorporates the powerful footwork and fast spins. Nritta is what is usually used as a way to show your skill. Nritta is what is usually performed for audiences that don’t know much about Kathak as it is exciting for them. Nritta is the “wow factor” for those who know Kathak well. Nritya is the more expressional side of Kathak. This incorporates the nine emotions (navarasa). Nritya is used to show story.
Both aspects of Kathak are very important. None of them are easy to do and both take practice. Kathak is an art of storytelling so Nritya is what is used. But to show certain emotions Nritta can also be used along with Nritya. For example: If I am showing anger I can use Nritta to show it as well by doing fast spins or footwork to emphasize which emotion I’m showing. If I am showing peace and tranquility I will do soft movements and nothing fast or forceful.

