I get this question a lot, and wanted to give a brief explanation. In short, no, they don’t. I’ll sometimes point users to Do Websites Need To Look Exactly The Same In Every Browser?, but of course more detail is usually necessary. So, here it is.
Why not? Because each browser handles displaying elements differently; often it isn’t the best use of your client’s money to give users the same pixel-for-pixel experience on every browser.
What’s important is something called graceful degradation, because it focuses on letting users access what’s important—the content—regardless of what their browser supports. The best browsers get to display the best features like drop shadows, rounded corners, sweet CSS-based animations and other goodies. Browsers that can’t support those still get the best experience they can afford; sometimes that means no rounded corners on your buttons (hi, Internet Explorer 7 and 8 users).
Of course, it’s possible to add these things after the fact with jQuery and the like, but I’ve found it really just makes things cumbersome when you go to update something a year or so down the road.
You also need to ask yourself if supporting an older browser is right for your site. Take a look at your Google Analytics reports (you do have that, right?) and see what percentage of your visitors use Internet Explorer, and what percentage of them use older browsers like Internet Explorer 7 (or 6 even). Depending on the size of your site, 3% may be the magic number for dropping IE6 support. Other sites it may be 15%. It really all depends on how many visits and page views you get per month from that browser.
It’s nice to know that since Microsoft, Chrome and Firefox (Apple, you out there?) are turning on auto-updates, since discussions like this will one day be irrelevant. Seriously, thank you, Redmond.
While it’s important to consider your users, you also need to make sure you’re using looking forward and using the best technology possible for your site. If you keep looking back to support older browsers, your site’s current version won’t stay as fresh as it could.