On developing websites, we try to create widely valid HTML, allowing the use of the pages on every system and with every browser. Unfortunately there are a lot of, especially older, browsers that do not care too much about standards, so even completely valid HTML might look "quite different" on those systems.
Nevertheless we dislike "code optimizations", because they make the source code bigger and less efficient, can slow down displaying and finally, even if you made a website look exactly the same on the systems you tested it, it may not look the same on a not tested system. So once you started...
As we believe that the Internet should be about contents first, we try to produce usable web pages - usable on nearly every system in the world. To provide this and backward compatibility, we test our projects with older browsers like Netscape 3.0.
Design is second, but still extremely important as it the main factor for the "first impression" of a site. That' the reason for testing all of our designs on several browsers: Currently we use Opera as our reference browser. If the design is OK there, Mozilla Firefox is used to check again. We do not use Microsoft Internet Explorer as a reference, as it still does not comply to many standards. Nevertheless, as most of the visitors of our customers use it, the design also has to be acceptable on this browser.
Besides focusing on the look of the pages on the screen, you may also ask for an improved print layout of the website.