Looking at screen-scraping at a simplified level, there are two primary stages involved: data discovery and data extraction. Data discovery deals with navigating a web site to arrive at the pages containing the data you want, and data extraction deals with actually pulling that data off of those pages. Generally when people think of screen-scraping they focus on the data extraction portion of the process, but my experience has been that data discovery is often the more difficult of the two.
The data discovery step in screen-scraping might be as simple as requesting a single URL. For example, you might just need to go to the home page of a site and extract out the latest news headlines. On the other side of the spectrum, data discovery may involve logging in to a web site, traversing a series of pages in order to get needed cookies, submitting a POST request on a search form, traversing through search results pages, and finally following all of the "details" links within the search results pages to get to the data you're actually after. In cases of the former a simple Perl script would often work just fine. For anything much more complex than that, though, a commercial screen-scraping tool can be an incredible time-saver. Especially for sites that require logging in, writing code to handle screen-scraping can be a nightmare when it comes to dealing with cookies and such.
In the data extraction phase you've already arrived at the page containing the data you're interested in, and you now need to pull it out of the HTML. Traditionally this has typically involved creating a series of regular expressions that match the pieces of the page you want (e.g., URL's and link titles). Regular expressions can be a bit complex to deal with, so most screen-scraping applications will hide these details from you, even though they may use regular expressions behind the scenes.
As an addendum, I should probably mention a third phase that is often ignored, and that is, what do you do with the data once you've extracted it? Common examples include writing the data to a CSV or XML file, or saving it to a database. In the case of a live web site you might even scrape the information and display it in the user's web browser in real-time. When shopping around for a screen-scraping tool you should make sure that it gives you the flexibility you need to work with the data once it's been extracted.