There are a couple of ways in which browsers differ from one another. One of these is the rendering engine that the browser uses to convert the HTML and CSS that define the web page into an actual displayed page. The other is the functionality that the browser provides or which can be added to the browser via plugins or extensions.
When it comes to the rendering engines that the browsers use there may be minor differences in the way that they render the page depending on just how compliant they are to the standards but basically the days where different browsers would render pages completely differently are now a distant memory. Those writing web pages may concern themselves with trying to make sure that all of the popular browsers display the page the same but for the most part the page will display satisfactorily in any modern web browser even if there are minor differences in the exact layout. Since the people actually visiting the web page will generally only be using the one browser they will be completely unaware of any differences in the way th looks in their browser compared to how the author intended it to look. Which browser someone chooses to use will seldom be decided based on the rendering engine the browser is built on.
Functionality is a completely different matter. If a web browser doesn't provide the functionality that someone expects while another browser does provide that functionality or provides a means to add that functionality then the person is very unlikely to choose the browser that doesn't have it.
Different people interact with the web in different ways and so the functionality they expect their browser to have so as to allow them to interact easily will differ from person to person. Browsers can deal with this in one of two ways. They can either build in all of the most popular functionality so as to ensure that the functionality works as efficiently as possible with the overhead of it being there even if some browser owners never use it or they can provide that functionality in the form of plugins or extensions where only those who want the functionality will install it but where as a separate module it will not be fully integrated into the browser and so will not work as efficiently or may interfere with other extensions or plugins.
So we have some browsers that have lots of functions actually built into the browser and others that provide far less functionality out of the box. Regardless of the amount of functionality built in most browsers these days provide for additional functionality via plugins or extensions. In some cases extensions become so popular that they get integrated into the browser itself. Less commonly, a browser drops some of its integrated functionality and then you have to use an extension to add it back in.
Apart from fixing security issues, the main difference between versions of the same browser is the additional functionality that has been built into the core of the browser. A new browser version may implement a newer version of the rendering engine but few people will even notice that. Even where a browser changes to a completely different rendering engine that is normally done without changing any of the functionality that was available in the previous version.
Different browsers have taken different approaches to how they provide functionality. Firefox was the first browser to provide an easy way to add extensions to add functionality that could be created in a simple way that didn't need the module to be compiled. Opera experimented with new functionality most of which got built into the browser and were many of the more popular options were quickly produced as extensions for the other browsers. Google Chrome took a minimalist approach where very little functionality is actually built in and what there is mostly needs to be accessed via the keyboard.
Having selected a browser most people stick with it for a long while as the upgrades generally don't make any difference to the way they use the browser. There may be additional security, an updated rendering engine, and a few new functions but the functionality that the person was using before the upgrade still works the same way after the upgrade. Of course some people do change browsers either because they find another browser that provides the same functionality and which runs faster or because their functionality requirements have changed and a different browser now meets their needs better.
People are only forced to make a decision about changing browsers if they are using one of the less popular browsers and those who produced the browser decide to cease support for it. If another browser exists that provides all of the functionality they need then they have a browser they can switch to that may perhaps be not quite as fast or as efficient as the browser they have been using but which at least provides the same functionality. The biggest problem comes when a browser is no longer supported and no other browser provides the functionality you need. Then you end up with the situation where you have to decide how long to stick with a browser that is no longer supported but which provides your desired functionality. Presumably the browser will never have been popular enough to be a serious target and so the lack of security updates will probably not be an issue. The lack of new functionality will also not be an issue as you are sticking with the browser precisely because it provides the functionality you need. It is at this point where the rendering engine will probably be the deciding factor. Those browsers that are still supported will get a series of minor tweaks to their rendering engines that will add support for new CSS3 commands and eventually all of the popular browsers will be able to display pages correctly that use those commands meaning that people will use them in their pages. The no longer supported browser will not receive the updates to render these correctly and so there will be a growing number of web pages that no longer display in the browser the way their authors intend. In some instances this will interfere with the functionality of the web page and a choice will need to be made between web page functionality and browser functionality.