QPushButton is simply a button. QToolButton is part of a group of widgets in the QtWidgets module that operate on QActions: QMenu and QToolBar are other examples. As a result, QToolButton is much more complex under the hood than QPushButton.
"A tool button is a special button that provides quick-access to specific commands or options. As opposed to a normal command button, a tool button usually doesn't show a text label, but shows an icon instead."
When i want a button in the GUI simple with only an icon, I use QToolButton. But when i want a classic button, i use QPushButton.