There are some situations when paid vs free has a meaning or not. Here are the differences:
PAID: Application can be installed only after it has been purchased with a supported payment method. This gives access to all the functions.
FREE: Application is free, but developer might implement additional features, that you have to buy to unlock it. Similar to PAID, but it's more like trial.
DONATION: You get the software for free and if you consider it useful, a payment can be redirected to the developer to support further development or support.
FREMIUM: The application is both FREE and PREMIUM, where developer sells additional features through the FREE application.