At the time this question was asked Firebug was a beast, but now the Chrome Developer Tools (DevTools) are handy for web developers. Though I'm not ranting about Firebug because I've learnt the web development using Firefox with Firebug.
It was a great tool for web development and it introduced all major features of DevTools and Firefox's DevTools has. However, I switched to the Chrome DevTools though they don't cover all the features of Firebug, because they are light-weighted and much faster than Firebug, accessing localStorage is easily defined and the sources are organized there in my opinion.
Here I'm going to list how the features are unique in Firebug,
Search:
Search option is well-defined in Firebug, because its easily accessible and we can search keyword with case-sensitive and regular expression.
DOM:
The DOM structure can be easily accessed in Firebug with various filtering options like Show User-defined Properties, Show User-defined Functions and so on.
Cookies:
Firebug lets us create our own cookies and gives provision to export cookies.
Network/Net:
Firebug has a Net panel wheras the DevTools call it Network. Both are useful to analyse all the requests made to load the resources and their status. In Firebug, we can easily grasp the resources' Remote IP.
Sources:
Even though the Source Edit is available in the DevTools, I feel Firebug is better while using Source Edit, because if you want to edit a CSS file within the DevTools, you have to go to Sources panel, then press Ctrl+O to find the file. Only then you can edit the file. In Firebug you can easily find the Source Edit under every menu tab.
Support for dojo:
Once I was a dojo developer Firebug was easily extended to support dojo developement by using the Dojo Firebug Extension.
