Stopping distribution means that Adobe will no longer host official Flash Player plugins for download at all, meaning that old Flash games that try to helpfully auto-install the Flash player for you will fail as those links will go dark. Presumably archive sites may preserve installers to manually install the Flash Player after the 2020 shutdown date, but that will be an increased amount of effort that will make it less likely people will do it.
Stopping updates particularly means security updates, which is an increasing reason for the ongoing maintenance of the Flash Player. That will add an increased reason for people to avoid manually installing the Flash Player as it will be an increasing security liability.
Most browsers are already on the path to disabling the sorts of plugin APIs that Flash Player has used for security footprint reasons alone. When the Flash Player no longer receives official security updates, the browsers will have increased reason to eliminate that risk, and disable those APIs for good. That means that a manual install of the Flash player will be increasingly unlikely to even work soon after the shutdown date. Most browsers have already been planning to disable such APIs and break such plugins near or around the date Adobe has given already, so this process has largely already started.
Flash content and old Flash games are going to break. The web will be a little bit different at that date.