WebParts represent SharePoint modules which can be used to extend pages of the platform.
A WebPart is a complete component that can be placed anywhere on your pages and is fully connected to your SharePoint platform. It can be developed to display, add or change data stored in your libraries and lists.
Add-Ons are extensions which are developed to simplify daily work with SharePoint.
An Add-On is a feature that extends native functionalities of SharePoint. It can be displayed as an additional button in places such as the SharePoint Ribbon Menu. There it can be easily accessed by users and can perform custom actions on items, files or lists and libraries.
Solutions are complex systems of features and functionalities created to modify or enrich your SharePoint platform.