Flask is a python microframework which depends on the Jinja template engine and the Werkzeug WSGI toolkit. The “micro” in microframework means Flask aims to keep the core simple but extensible.Flask won’t make many decisions for you. Flask supports extensions to add functionality to your application as if it was implemented in Flask itself. Flask may be “micro”, but it’s ready for production use on a variety of needs.
Quart is a Python web microframework based on AsyncIO. It is intended to provide the easiest way to use asyncio in a web context, especially with exiting Flask apps. If you are new to Python you should start by reading Installation, if you are new to Quart then see Quickstart and if you are also new to AsyncIO see AsyncIO. If however you are very familiar with Flask, see Migration from Flask.
Quart is an evolution of the Flask API to work with AsyncIO and to provide a number of features not present or possible in Flask, see Flask evolution. Compatibility with the Flask API is however the main aim, which means that the Flask documentation is an additional useful source of help.