You can turn your firewall back on. I'm using Windows Firewall and it's probably the least robust or all firewalls. It's VERY unlikely to be your problem, though I've seen one person say that was it. But if it's not working with it off, either, DON'T leave it off. Unfortunately, people who can get here and the homepage still occassionally have activation problems, so the two aren't related. Usually, though, it probably works for 90% or better.
Try running the setup as administrator. If it's working in an XP environment and not in Vista, that's a likely scenario, although the problems that doing that solves are usually different than not activating. Still, I'd try that first. Is this the first GOTD you've tried? If so, you're browser might be setup to run through a proxy server, as our on top of things moderator, Bubby, has pointed out in the past. If you're not aware, a proxy server is used in a business/institutional environment, rarely at home, and a lot of people aren't aware that they have the option in IE enabled. I'm not even sure that isn't the default. Go to the Tools menu, then Internet Options>Connections tab>Local Area Networks (LAN) settings and click on the button. Then uncheck the proxy server bubble. That has given problems to more than one GOTD user. This only applies if your not using dial-up.
If you have a backup dial-up account, you could try that. If it works, then you'll know it's a network settings problem, like the proxy server setting. While you're in you're internet options, click on the programs tab, and the manage addons button. You can check and see if you have newer version of Java. That's not a prerequisite, though since GOTD uses some of scripting, it's helped a person or two. You can see the java add-on in one or two of those list. I believe it shows under the "without permissions" option on the drop down menu, but my IE is currently damaged and I can't check. =S
http://www.java.com