Old topic - but an important aspect not covered here is that a "renewal" can generally be actioned within a 90 day window - from 60 days prior to expiry to 30 days after expiry. Ref: https://ie.godaddy.com/help/renewing-my-ssl-certificate-864
A "rekey" on the other hand requires the certificate to be updated within 72 hours.
The question to ask is "are we concerned that the key may have been compromised?". If the answer is "No" then a renewal requires less time and is easier to coordinate, especially when involving multiple parties.