I would like to install some Firefox add-ons, also called 'extensions' or 'plugins', like for example Adblock Plus, Ubufox or Greasemonkey.
Should I install them from the Mozilla website, using the Firefox browser, or should I search Firefox in the Software Center and then select the interesting applications ?
I suppose the issue is about how the extensions are kept up-to-date. If your answer is 'via the Software Center', how can I migrate well from my former add-ons without losing all my preferences?
I would prefer Firefox's inbuilt add-on manager as it contains all add-ons of Firefox and will get frequent updates.
I'm going to say both, it depends on the add on.
If you want to keep on the outside of the envelope, use the built-in add on manager. Add ons installed this way will be updated frequently, and you will get major version updates.
If you prefer a bit of stability, use the ones from the repositories. They are (mostly) very heavily tested and pretty much guaranteed to work really well.
Add Ons installed from the repositories are not updated between major versions, they only get security and bugfix updates. This is a big plus for stability.
The ones provided by canonical are also eligible for canonical's enterprise support, if you're interested in that sort of thing.
Needless to say, anything that is operating-system dependent (e.g. ubufox) should be installed from the repositories. The same goes for security-related add ons (even though the only one I can think of is pwdhash). There's no harm (at all) in installing all else via the add on manager.
If you just remove the items from the software centre, your preferences should be kept. But the add on needs to be aware of previous settings, so it may not always work.
Mozilla reviews and test every extension public available and approved, which is the case of all extensions in the package manager. So you are pretty safe using Firefox add-ons manager or even downloading from Mozilla add-ons site, as long as you don't download unapproved extensions.
Downloading from Mozilla has the benefit of getting the most up-to-date versions, which is particularly important for extensions that needs to be updated frequently. For instance I had to release a new version of one of my extensions 10 days ago, because it stopped working due changes on YouTube web site. Six days later, Vimeo changed their web site too and the extension stopped working again, so now I'm rewriting the extension. This doesn't happen very often, but when it does, you get stuck with a non-working extension, if you can't get an update right away. To be honest, I don't know how often extensions are updated in Ubuntu package manager, but I suppose not much, since they follow the same update policy for other Ubuntu applications.
The advantage of the package manager would be stability, but if something goes wrong with an extension update from Mozilla, you can always go to the extension page, add /versions to the url and download the last stable version.
I used to have more than 60 extensions installed in Firefox, and except for Ubufox, all of them were downloaded from Mozilla.
Bottom-line, in my personal opinion, you should get your extensions from Mozilla site.
BTW, add-ons is a generic term used by Mozilla to describe extensions, plugins and themes. But there is a big difference between "extensions" and "plugins". What I'm talking about here are extensions, since plugins are not hosted on Mozilla site. They have a page about plugins and the browser has a functionality to check for plugin versions, but the update process is done via developer site. The automatic plugin installation, that is presented when the browser detects you don't have the necessary plugin for a specific content, does not work on Ubuntu and the plugins must be installed manually or via package manager, which is the most common on Ubuntu.