Is the Zotero web library the same as the Zotero desktop app?

The Zotero desktop app is the primary way of using Zotero and offers complete functionality. When people talk about “Zotero”, they're almost always referring to the desktop app.

The Zotero web library is a complementary tool for accessing your Zotero data when you're away from your main computer or using a platform that can't run Zotero (locked-down institutional computers, some Chromebooks, etc.). The web library also allows you to share a Zotero library publicly with people who might not use Zotero.

A desktop app allows for vastly more functionality than a website. Here's an incomplete list of features that only the Zotero desktop app provides: