Tooltips are most commonly used on hover to indicate what clicking on an icon or link will do; this way a user can "investigate" the button and know what it does before clicking. As Microsoft's guidelines put it, "Tips help users understand unknown or unfamiliar objects that aren't described directly in the user interface (UI)." It's not a good way to express more general information, a tooltip should refer only to explain how to use a single element of the interface.
Tool tips also have some other downsides that make them less than ideal (See "the downside of tooltips"), longer ones in particular are hard to read, and keyboard or screen reader users may miss them as they won't "hover." More importantly your users probably expect to know exactly how to use an email field, so they won't look for a tooltip.
On the other hand the confirmation dialog is pretty perfect for this role. A confirmation dialog has the advantage of calling the user's attention to what they have to do now; the tooltip in this case will tell them what they have to do in the future, if they read it. The confirmation is after the form, so it's simply telling them what to do as a next step, and you know it will be presented so you avoid the pitfalls of the tooltip.
As an added bonus, you can set up your system to only display the confirmation message about the confirmation email when it's necessary. If a user doesn't change their email they'll never see it.
