There are generally two flavors of email for the average person. IMAP and POP.
IMAP lets you sync your email on all devices that you use that email address on. Delete an email from one device, send from another and ALL devices will reflect those changes. Because the email is actually stored on your account provider's servers.
POP email can be used on multiple devices, but each device is an island to itself and operates on a first come first serve basis. If you download email on a POP account to a Mac say, your iPhone will never see it because it's already been downloaded to your Mac and removed from the email provider's server. Same with Sent items, etc. Email is stored on the device it downloads to or originates from.
Exchange is a little different because it's your work's servers doing the storing of your email.
Since you added your Gmail account that way it could be POP. I don't know. For myself, I generally ignore all that stuff and just directly enter my settings (it's all in my head). I use IMAP with all my accounts.
If it is POP, it's possible it's storing something somewhere on one device and not on the other.
