It's a problem when using email to register into a different site. Services without email validation will happily accept those two addresses separately, because they don't adhere to Gmail's internal conventions. If the site then sends mail to its registered accounts, then the Gmail address will receive unwanted mail.
There are horrible people who loves the idea of being able to use someone else's identities. Not just made up but stolen. I - I don't want to know why, nor what they're normally thinking.
Will Gmail prevent you from entering such an address elsewhere, thinking either that it is yours, or that it is the email of some intended recipient other than the actual Gmail accountholder?
Alternatively: will third-party systems consolidate such addresses under a single account, or treat them as different?
You have to be a bit careful with that kind of argument though. According to the specs the local part of an email address (before the @) can also be case sensitive so me@example.com and ME@example.com could be different email addresses. In the early days of a new system we implemented account email addresses following what the rules really say and making no assumptions. We then spent way too much time dealing with users who signed up with one address but then tried to log in with a "different" one because they had caps lock on or some auto-capitalisation feature in whatever browser they were using. Lesson learned - users do assume those are all the same address and cleaning up a database that sometimes has accounts for both me@example.com and ME@example.com isn't much fun.
There is no RFC about how the local-part is interpreted: even the gmail +tag syntax is an extension some servers implement and some don't. The only way to accurately test equality of email addresses is to send an email to them.
Even that only tells you something about equality/equivalency (or more precisely, the deliverability) at this moment. I have a family mail server that we periodically update mail list and forwarding rules. You could test for deliverability today and tomorrow the situation could change.