Renewing an SSL certificate is the functional equivalent of generating a new certificate; it is not a revocation, and the 'old' certificate will continue to be accepted as valid, assuming it has not yet expired. There is no downtime/maintenance window requirement for its replacement.
Note however that due to the SSL SHA1 Hashing Deprecation, many certificate authorities (CAs) are moving quickly to phase out SHA1 in favor SHA256 by the end of 2016. This means that many CAs are making it easy to "re-key" existing SHA1-based SSL certificates, providing a quick regeneration of the existing SSL certificate, using the new SHA256 hashing algorithm, usually with a few clicks. GoDaddy, a major CA/domain registrar, implements this feature via their Starfield SSL CA service.
What is not often made clear, is that a re-key of a SHA1 cert to a SHA2 cert can include an implicit revocation of the existing certificate. The underlying process at Starfield to rekey a SHA1 certificate generates a revocation request, immediately rendering your current SSL certificate invalid. There is no documentation or warning on this important detail (!). Many will assume requesting a re-keyed certificate is the equivalent of of a certificate renewal, when in fact it is a revocation request, plus the generation of the new certificate. Imagine your surprise after clicking "re-key" and then downloading the new certificate, perhaps waiting for a later install, and that within a few hours your SSL endpoint become invalid. Not fun. So be very careful on re-keying SHA1 certificates during the SHA1 deprecation transition period.
Another point to be aware of when migrating from SHA1 to SHA2 certificates, is to make sure and download and install the new CA intermediate certificate chain into your keystore at the same time. The SHA1 to SHA256 migration often means that the new SHA256 certificate requires new intermediate SHA256 certificate chains in your key store. Not doing this can result in the invalidation of the base certificate.
