Esri currently offers 3 different web API's that can be freely downloaded.
Are they all equal?
If not, what are the relative strengths/weaknesses of each of these API's?
From a totally abstract management perspective, the three APIs are equal. They represent code that runs in a web browser, the purpose of which is to display map data to a user over the internet. You can create a successful, meaningful mapping application with any of the APIs.
Furthermore, the impact of each strength/weakness (difference) in each API will vary depending on the audience. Programmers will be keenly interested in language features or drawbacks, your network folks will want to know bandwidth requirements and server prerequisites, the GIS folks will be deeply concerned that the displayed map is truly rendered, and your end users could care less about all of that and just want to do whatever it is they started doing.
So here are some key items about the three APIs:
The Silverlight and Flex API's are by far the most popular ArcGIS API's and have the advantage of being browser plug-in's which provide a much richer and consistent user experience which is often quite important with GIS web apps. Silverlight is a probably the best choice if you favor .NET development and Flex would be best for developers that prefer the Adobe environment. Flex also has the advantage of wider adoption as a browser plug-in than Silverlight.
There is an ArcGIS Viewer for Flex.
There is an ArcGIS Viewer for Silverlight.
See related question.
"Viewer" is a term Esri uses for web apps that allow a user to create mapping web apps without writing any code. Typically these created web mapping apps will be viewers, but with the appropriate add-ins (or widgets) it is possible to create viewers with editing capabilities.