Use 'pay out' when referring to a large sum of money being paid, often as a reward or settlement.

Large Payments

'Pay out' can imply reluctance or obligation to give money, not always done voluntarily.

Not Always Willing

In financial contexts, 'pay out' often relates to dividends or insurance claims.

In shipping or climbing, 'pay out' is about controlling the release of a line or rope.

When paying out a line or cable, the action is gradual to maintain safety and control.

The nautical use of 'pay out' is often heard in maritime and nautical contexts.