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For non-GIS background devs creating single-function mapping apps - I'd probably recommend Leaflet (now supported by MapBox). Easy to use and small. More functionality relies on plugins of varying quality and support.
GIS online type application - use OpenLayers3 - has the full suite of data sources, controls etc. in a single library. It can also be used for simple mapping apps, so if you have a mix then I'd use this for everything.
Google still has the library/data combination (e.g. StreetView is not available anywhere else).
Esri has GUI web app builders, and as noted below, why complicate things if you already use their stack.
Commercial APIs (Google, Bing, Yahoo)
Using any commercial API leaves you at the mercy of whatever changes the provider makes to the API or Terms of Service. What happens for example if suddenly your local government portal that uses Google Maps suddenly has adverts popping up all over it? Want to reuse your JavaScript Google Maps code for an Intranet site? You'll find yourself having to pay the $10,000 licencing fee.
Microsoft often have more defined and fixed terms for their services (if paying) so there may be less of a risk here.
The recent backlash against Twitter is a good example of developers having an API that changed beneath them. If you aren't paying for a service then you (or your system) are the one being sold.
Google started charging for use of their maps from October 2011.
Esri
As others have mentioned if you are using an Esri stack then the ArcGIS API for JavaScript will no doubt work well on top of it. Coming from a GIS background Esri have probably thought more about traditional GIS tasks and features than the "neo-geography" web-giants (though this is just an opinion/feeling).
OpenLayers has built in support for ArcGIS REST layers, and if you are looking to reuse your code for non-ESRI based websites then again an open API serves you best.
Use OpenLayers...
I can't really think why developers would use an API other than OpenLayers. Open Source projects lead to related Open Source projects so there are a wealth of reusable components out there such as the GeoExt library, MapQuery, and GeoPrisma.
I'll just add that just because a project is Open Source does not automatically make it better than commercial equivalents - but the OpenLayers API matches the commercial competition in this case, and the ability to see how the source works, the unit tests, the build scripts etc. mean that you can easily build new features on top of it.
There has been some recent criticism of OpenLayers, mainly related to complexity, styling, and size. There have been counter-arguments made to these here and here by Christopher Schmidt one of the lead developers of OpenLayers.
It is worth noting that if you need a simple Open Source mapping API then take a look at Cloudmade's Leaflet.
The comment smaller, faster, newer, and more straightforward can also be read as less features and less tested.
Scan the API documentation for Leaflet and OpenLayers. The latter includes items such as WFS layers, editing tools, and SLD support. It has also been tested in many different environments, and works in IE6 (allowing government and local authority users).
For simpler display of spatial data Leaflet looks ideal, and easier to get started. However I'll be sticking with OpenLayers for more feature-rich GIS applications.
Caveats
One possible downside is often new innovations are seen first in commercial providers' systems - however these nearly always filter through to OpenLayers in time.
Finally I'm sure there are certain scenarios where other APIs are more suitable - on custom hardware, to fit in with an organisations other IT systems, or if you already know an API inside-out and can get a system developed in half the time.
All the APIs you mentioned are capable of producing great online mapping systems, but your choice should also fit in with you or your company's future development needs.
Best Answer
For just drawing vectors, Leaflet or OpenLayers will do just fine. Nothing new there.
For vector tiled maps (i.e different resolutions/levels of detail at different zoom levels), TileStache is a simple and good server. I have been using with lots of experiments with Polymaps (AFAIK, the only general-purpose working js tiled vector viewer). Some people have used OpenLayers for adding tiled vector support, but AFAIK, it has not been integrated to trunk in a year, so it probably won't in the near future for lack of interest.
TileStache (the server) itself is working very well. Nevertheless, I have found several bugs with Polymaps and since it is not being updated anymore (abandoned project), I cannot convince myself to use it in production (I'd have to maintain it myself... I can't justify it to myself just yet.)
My solution for me has been to write a custom OpenGL native vector renderer which is used in AmigoCloud. In fact, that is how all maps are rendered in our service. For that purpose, TileStache has delivered quite well.
You want to look at the TileStache Vector Provider. For examples on how to use it, you can look at a very basic vector test suite I wrote.
Sadly, I don't think Leaflet has support for tiled vectors.