Choose Zotero for Firefox if:
Zotero for Firefox runs as a plugin for the Firefox web browser, not as a separate program.
The same version of Zotero for Firefox works on any computer that runs Firefox; there are no separate Windows and Mac versions.
Choose Zotero Standalone if:
There are different versions of the Zotero Standalone client if you are using Windows, Mac, or Linux and you will need to install the right browser extension for each browser you use.
Installing the Zotero Standalone client requires downloading and install a program for your operating system (Windows or Mac) and then installing browser extensions or add-ons to allow the client to talk to the browser.