When it comes to creating web pages, content is king (as the saying goes). This means that those creating web pages need to ensure that they have content that people will want to read and that can be accessed from whatever browser their visitors choose to use. Web applications are a bit different as there the actual functioning of the site is also relevant.
When it comes to individuals choosing which browser they will use to access the web none of the above is really all that relevant (it is relevant to the web page creator ensuring that their page will be usable on as many browsers as possible but it isn't really relevant to the web browser user).
When choosing a browser functionality is just about the entire story. You will choose to use a browser where the browser provides all of the functionality that you want to use in the browser. Since Netscape 4 died the core functioning in just about all browsers has been the same (IE6 through IE8 used different commands for some things but almost all sites were written to cater for that). The main exception to this is where sites have deliberately set out to exclude specific browsers from being allowed to work with their site. Since in just about all cases there are alternatives to sites that do this that can be used instead all that these sites achieve is to lose potential visitors. Anyway, in most cases this can be overridden so as to get the site to work with your browser if you feel that you really need to use that particular site.
Apart from those who use a particular browser because they don't realise that alternatives exist, most people will choose which browser to use based on something beyond the core functionality of the browser. There are a number of different areas where these non-core differences might be depending on just what is most important to you.
For some people the choice of browsers depends on the extra options it provides that you can't get with the other browsers (or where the other browsers only offer it via a plugin or extension). A lot of people selected Opera for this reason as for many years it was well known for being the first browser to introduce new features - many of which were eventually copied by the other browsers. Note however that in many cases it isn't so much whether the different browsers have a given feature or not but how easily it can be found when you are first looking at the browser.
The availability of specific extensions or plugins may be another reason for choosing a particular browser. If no browser has all the features you want built in and you can get some of the extras on one browser using extensions or plugins then perhaps that will be the reason for choosing a particular browser.
Whichever reason you have for choosing the particular browser, you will probably continue to use that browser (and its upgrades) for a long time without considering the alternatives. This means that there may be an upgrade to one of the other browsers that actually introduces all of the features you want that would make that browser a closer match to your requirements than the one you are using.
People choose to use a particular browser for any of these reasons (and others) and the likelihood of a particular web site being able to convince them to switch browsers is almost non-existent.