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:
A local database fully under your control, with optional syncing
Fast, offline access to all your data — whereas the web library can only load a subset of your data at a time, the desktop app allows you to access and modify everything much faster
Word, LibreOffice, and Google Docs plugins that let you insert citations and bibliographies directly from Zotero and keep them updated automatically
A better experience when saving items from the Zotero Connector (+
other features)
The ability to work with a large number of references — certain operations, like export and bibliography generation, are limited to 100 items in the web library for technical reasons
More comprehensive export output with options to include files and notes in some formats
Real windows that you can interact with via Cmd-Tab/Alt-Tab, resize, etc. — not just an interface within a single browser tab
Local filesystem access — e.g., adding PDFs directly from your documents or creating links to files on the local disk
Automatic retrieval of metadata for PDFs
Find Available PDF to locate open-access versions of files
Importing of references from a file or from other reference managers
PDF full-text indexing (coming soon to web library)
Advanced search and saved searches
Retracted item notifications
My Publications collection to share your work publicly
CSL style editor
Custom citation styles
Unfiled items collection (coming soon to web library)
Duplicate items collection and item merging
Ability to create related items
A user interface that adapts to the
OS you're using
An unrestricted plugin system that allows for many more features and further customization