In the beginning the choice was clear
Desktop PC programs (viz. Windows Programs, Native Apps) were all we had at the dawn of PC programming in the early 1980s and Cambria was there. The program was typically written in a language like dBase or Turbo Pascal and stored right on the computer (or installed on the computer if it was purchased). The data it used and stored, if any, was also stored in a database residing on the PC. If your computer crashed and you did not have a backup stored somehow you were out of luck.
The advent of the World Wide Web in 1991 changed everything. Before long a program could be stored on a remote computer (a “server”) and accessed and run using a browser. This “Web Program” had many advantages that quickly made it popular. Gone were hassles about installing the program on your computer, configuring it and the database and keeping it updated to the latest version.
Fast forward to today
The boundaries are now not so clear-cut. The advent of automatic updates eased made having an updated desktop less of an issue. Moreover crashes were not such a catastrophe. A desktop app can now store data on the local computer while automatically backing it up on a remote server. Or the data could be on the remote server “in the cloud” and let the professionals worry about safety and backup.
So the choices now requires some thought. Sometimes a desktop choice is easy. If you never use the software except on your local computer and you appreciate the slicker interfaces and speed of response of a local computer then a desktop application can be exactly the right choice. Today many famous programs are desktop apps. Examples include Skype, Macromedia Dreamweaver and Spybot Search and Destroy.
Other times a web program is the obvious way to go. An example would be a worldwide company wishing to have software that everyone can use and accessing a common database. The subject has yet to come up on this page but increasingly programs are run on mobile devices, usually using software specially written for the small interface and in case you are a worker in the field, without a desk to work at, the choice is clear.