Desktop application are targeted for specific platforms.
Web application add one more layer between user and server i.e. browser.
Mobile application further add complexity by addition of devices types, Operating System and faster update and upgrade cycle. Limited by size, memory and processing power and battery life.
In web application, we test for energy efficiency for only few devices (Web Servers) where application is hosted. While in mobile scenario energy requirement for clients is main concerns. In Desktop application, performance testing involves finding memory and processing power used at client machine.
Desktop and web application have only two inputs mouse and keyboard while mobile have multiple ways of interaction like gesture control, voice commands, GPS locations, tilt, movement, eye direction, sound (Google Assistant, Cortana, Siri) etc.
Browser Compatibility Test, web accessibility testing,
Performance and security testing are different in web and desktop application.
In web testing, performance is measured by loading time of web page and resources consumed by server in processing user requests while in mobile performance is measured by battery consumption, memory requirements.
With advancement from Desktop to web to Mobile applications, the complexity of testing is increasing at exponential rate.