The Web is Not Print
Many Web designers with a print background come to the Web. Either they were print designers, or they are just used to the control that a print world gives. When you print something, it provides permanence and stability. You don't have this on the Web.
The problem is, that it's easy to forget. When you build your Web page and test it in your browser, you get it looking exactly how you want it to look. But it looks different when you test it in a different browser. And it will look differently again if you move to a different platform.
You'll need to work with customers as a web designer. Especially if you bring your portfolio as print outs, you will be doing them and yourself a disservice if you don't explain the difference between print and the web page upfront. This is a common issue, where the customer demands the printout to represent exactly what the page will look like.
What Should You Do?
Working with Customers
Know what browser your client uses
If you're a big Firefox fan, and your client only uses Internet Explorer for Windows, you should keep this in mind in your designs. Your page could look very different to them.
Be upfront with your customers. If they want their pages to have very specific layout, font, and design elements, be sure to explain the tradeoffs such as download speed and maintenance before simply building them a completely graphical page.
Printouts as a Portfolio
It is always important to have a portfolio, but remember that the web is not print, and bringing a print out is not a strong representation of your website design skills.
Test test test
Test your designs in every browser and OS combination that you can get your hands on. Emulators work if you have no other choice, but there is no substitute for hands on experience.
Know your audience
If you design a site that suits your audience, your customer won't be complaining to you later. Know the characteristics of the audience of the site you're building. If they are propellor-heads, they might browse in Unix on a 21 inch monitor. Or if they are more conservative they might have a 12 inch monitor running Internet Explorer 3.
Don't forget screen resolution
Browsers and OS are important, but if your readers and customers are browsing on a smaller screen than you design on, they could be unpleasantly surprised.
The Web is Not Print
While it is possible, with CSS, to get very precise layouts, but it will never be as precise as print. If you can remember that as you're designing your web pages, you'll save yourself a lot of stress.