- Qt Widgets is a traditional desktop-app-looking UI toolkit, which can integrate with host operating system widgets and is mainly used through C++, Python or the QML language through https://www.kdab.com/declarative-widgets/.
Examples of apps made with Qt Widgets include: Guitar Pro, Unity3D bug reporter, Dolphin (both the emulator and the file manager), Calibre, Krita, VLC, VirtualBox, QTractor, LMMS, Mixx, and many KDE apps ...
Qt Widgets comes with a form designer (called Qt Designer) which generates XML files which can either be compiled to C++ (and basically just new's and set properties on widgets) or parsed at run-time.
- QtQuick is a fancy-looking, GPU-based, scene-graph backed, UI toolkit. It cannot integrate easily with, say, Win32 or Cocoa widgets because everything is rendered on a GL or D3D canvas if possible - as such, it is closer to game engines in terms of rendering than Qt Widgets. However, it supports fluid animations, and fancy effects akin to what you see on mobile or web apps.
It is mainly used through the QML language, but some classes have C++ bindings (and you can create your own QtQuick items in C++).
Examples of apps made with QtQuick include Substance Painter, Blizzard launcher, Native Access, Microsoft OneDrive (the desktop one AFAIK)... and it is used by the Jolla OS and plenty of embedded systems.
Qt Quick also comes with a designer (called Qt Quick Designer) - however in Qt Quick the code is much more integrated with the UI layout.