The payment date is purely administrative. The stock already adjusted on the ex-dividend-date, so nothing happens to the price when the cash lands. For a drip holder this is the day new shares are purchased and credited.
The gap between ex-date and pay date is why dividend income lags: you are entitled from the ex-date but funded weeks later, which matters for anyone running a dividend-capture schedule.
Example: ex-date 20 March, record date 20 March, payment date 15 April. A holder of 1,000 shares at $0.62 sees $620 in cash on 15 April, 26 days after the price already dropped by roughly that amount.
Related: record-date, ex-dividend-date, drip, dividend-capture