Hi,
Great question, because it is an important distinction to make. I'll review the Guide/API doc on this to see where we can clarify.
Callouts:
A callout is a UI element that can be used to display some useful information on a map. The information can be related to a specific point on the map, or it can be related to the location display. It can also be about a geo element like a graphic or a feature that’s displayed on the map.
Popups:
Popups are part of the ArcGIS information model and can be configured for individual layers (or sublayers) in a map via a popup definition. They can also be used with graphics and graphics overlays, and pixels and rasters. Popups determine how to display the data of a feature, graphic or pixel (geo elements) in a way that is easier to understand than basic attributes in a row/column format. They enable you to convert raw data into an information display that is both powerful and easy to read.
For example, when the user clicks on the MapView you could perform an Identify operation and use the Popup returned for the identified GeoElement as the content for a Callout.
On the shorter-term roadmap is a popup manager that provides helper methods and properties to make it easier for you to build your own UI for Popups.
For the slightly-longer-than-short-term roadmap we're also considering adding a standard Popup UI to the open source Toolkit that you could use out of the box or customize for your needs.
Cheers
Mike