in Zen Buddhism what is the difference between koan & laygroups?

A koan is a short, puzzling story used by Zen masters to help students move beyond discursive thinking. (Example below.)

The term "laygroups" has no meaning in Zen Buddhism. Are you referring to lay practitioners? If so, these would be non-monastic students of Zen. In the West, most Zen practitioners are lay people - not monks or nuns.

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Example of a koan:

A monk once asked Zen Master Chao-chou, "I have just entered the monastery. Please teach me, Master."

Chao-chou said, "Have you had breakfast?"

"Yes, I have," replied the monk.

"Then," said Chao-chou, "wash your bowls."

The monk was enlightened.

1. What did the monk attain?

2. If you were the monk, what could you say to Chao-chou?

Zen is a Japanese term, borrowed from the Chinese "Chan", meaning "to sit". It describes methodology rather than philosophy. Buddhism traveled from India, through the Himalayas, into China and then into Japan. Along the way it adopted local flavors and customs. So Tibetan Buddhism, headed by the Dali Lama, doesn't look much like Nichiren Zen from Japan. The basic tenets that Buddha taught, the four noble truths, the eightfold path, the illusion of samasara, of separateness, vs. the reality of nirvana, oneness, the need for each individual to explore and question all the teachings for his or her self, are common to most branches and schools, however.