This depends on whether the new plan is higher or lower in price.

Switching to a lower-priced plan

There will be no refunds for the difference.
The cost for the new plan will be charged on the new plan's start date.
(You can continue to use your current plan's benefits until its expiration date.)

Switching to a higher-priced plan

This depends on your payment method.

For website payments (Pastela Cloud)

You'll be billed the difference at the time of the change.
*The difference amount is based on a full month's cost, not a daily rate.

For in-app purchases

You'll be billed the cost for the new plan right away.
The Apple App Store will issue a refund based on the unused days of your previous plan.