Apple's Mail showing "the identity of smtp.gmail.com cannot be verified"

Solution 1:

In this case (April 4th, 2015) you could safely click "Connect". But in general such warnings should not be ignored. Here's how you can investigate future occurrences of such warnings:

Clicking "Show Certificate" and then selecting "Google Internet Authority G2" showed for this incident:

Google Internet Authority G2
Intermediate certificate authority
Expired: Saturday 4 April 2015 17:15:55 Central European Summer Time
This certificate has expired

And for "smtp.gmail.com":

smtp.gmail.com
Issued by: Google Internet Authority G2
Expires: Thursday 31 December 2015 1:00:00 Central European Standard Time
This certificate has an invalid issuer

So, the certificate for Gmail was still good, but the "intermediate issuer" that was used to create it didn't last as long as Gmail's certificate. That was an error at Google's; meanwhile they have installed a new certificate on smtp.gmail.com which uses a different issuer certificate. However, as this was trusted until a few hours before the problem started in April 2015 (and assuming you used it before, when all was good), it was safe to select "Connect" then.

Solution 2:

Google sent an email to those subscribed to alerts:

Google Apps for Business

Status: Service disruption

We expect to resolve the problem affecting a majority of users of Gmail at April 4, 2015 1:00:00 PM PDT. Please note that this time frame is an estimate and may change.

smtp.gmail.com is displaying an invalid certificate.

April 4, 2015 11:58:00 AM PDT