The Browser - Computing Model
With a Browser based application you basically have a program that starts and ends with current page displayed on the browser. Apart from a few things like cookies the current page has no way of knowing what has been going on in other pages.
For web sites that are online brochures or do simple step by step processes like a simple shopping cart this can be made to work tolerably well.
The Desktop - Computing Model
Unlike a Browser based application, a desktop application starts, retains as much information and resource as is requires until you say close.
Processing and editing the message on the right is something that should clearly be a desktop application.
80% Working is good enough!
The web world has introduced the concept that a web site/program that works about 80% of the time is good enough.
This was not an objective, it is a combination of people chasing fancy special effects, not understanding what they are doing, and sometimes the web site owner simply not being prepared to pay for a proper job to be done.
More details on this theme.
Interaction Between Controls
Many people will have tried the car insurance comparison sites, where a user selects the car manufacturer, make and model. This is a very large list and it is not really a good idea to load everything into the browser/desktop application.
Both application types would show a manufacturer selection list, when something is selected, the make list is populated and then the model list. In a browser app, this can be done by completely refreshing the whole page, which looks naff to the user, or by using an Ajax wrapper around the make and model controls.
The problem is that Ajax is unreliable in the Real World, so do you go for a good looking web site that may not work or a bad looking site that will.
The desktop application does not have this problem because it is more tightly bound together.
In practice the code required to get the data from the database will be same, but the code to update controls in the desktop application will usually be a lot quicker to write and test.
ASP/SQL Server/Windows or PHP/MySQL/Linux!
Although there are plenty of other options, it is most likely that if you ask someone to build you a web site you will be offered one of these two options.
It is possible to build a C# (ASP) web site on Linux/Mono/MySQL or PHP/MySQL on Windows, but if you do, you are going to raise a lot of eyebrows and find support difficult.