Ubuntu – Flash not supported on webapps

flashplugin-installer is the 'new' name for flashplugin-nonfree. The aim being to more accurately reflect the contents of the package - the package doesn't contain the actual flash player, only an installer that automatically downloads and installs a copy of the plugin. (The file downloaded is the tarball as provided on the Adobe website, but in this case the file is mirrored on Canonical servers.)

The reasons you still see flashplugin-nonfree in Synaptic is for backwards compatibility. It is a transitional metapackage with no actual content, all it does is depends on and install flashplugin-installer.

So to cut a long story short, there is absolutely no difference between the flashplugin-installer and flashplugin-nonfree packages.

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As for the adobe-flashplugin package - this is provided in the Canonical Partners repository. The difference with this package is that it actually contains the non-free flashplugin itself (it's not just an installer). Note that this package is exactly the same file as can be installed by downloading from the Adobe website.

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So I'm afraid I can't say definitively whether you'll see any performance difference between the flashplugin-* packages and the adobe-flashplugin - but at the end of the day they are the same plugin so it's very unlikely.