What are the advantages of an API connection vs an iCal connection?

Instant availability and supported rate updates are on the Plus side for the API connection.

Although iCal connections enable calendar synchronization between systems, it is important to note that this technology was not specifically developed for hospitality applications, and it isn't exclusive for this end.

iCal API
The synchronization of events does not occur instantly: High risk of overbookings. Immediate updates between software systems greatly reduce the risk of double bookings.
Rates aren't updated. Allows to control Rates.
Remote calendars are only updated when the OTA initiates a 'pull' of new updates, which could be as infrequent as once a day. You can swiftly and effectively push the latest prices to all websites: maximize your earnings during peak days.

API connection offers real-time connections; more supported features as the flow of data permitted is wider and the risk of overbookings is much lower.