To cancel or to downgrade depends on a couple of things

1) Is the card churnable

If the card allows you to obtain the bonus by applying again later, you should cancel. ( So you can apply again later )

2) Will you need to keep the credit line open for the score / reconsiderations

If your total credit line is not high, you might want to keep the credit line open to keep your credit usage ratio low ( which keeps your credit ) score high

Also, for issuers like Chase, when your application is pending or initially denied, you can request to trade the old card for the new card, and have the application approved that way.

In these cases, you should downgrade instead.

3) To keep your points

e.g. If you have earned Chase UR points / AMEX Membership Rewards Points where you haven't or don't want to redeem yet, you could downgrade your card to a no fee / lower fee card to keep the points.

( Although you will need to open a premium card in the future to transfer the points to FF programs )