The SSL certificate solves two purposes: encryption of traffic (for RSA key exchange, at least) and verification of trust. As you know, you can encrypt traffic with (or without, if we're talking SSL 3.0 or TLS) any self-signed certificate. But trust is accomplished through a chain of certificates. I don't know you, but I do trust verisign (or at least Microsoft does, because they've been paid lots of money to get it installed in their operating systems by default), and since Verisign trusts you, then I trust you too. As a result, there's no scary warning when I go to such an SSL page in my Web browser because somebody that I trust has said you are who you are.
Generally, the more expensive the certificate, the more investigating that the issuing certificate authority does. So for the Extended Validation certificates, the requesters have to submit more documents to prove that they are who they say they are, and in return they get a bright, happy green bar in modern Web browsers (I think Safari doesn't do anything with it quite yet).
Finally, some companies go with the big boys like Verisign purely for the brand name alone; they know that their customers have at least heard of Verisign and so that for people shopping on their online store, their seal looks a little less sketch-ball than, say, GoDaddy's.
If the branding is not important to you or if your site is not prone to phishing attacks, then the cheapest SSL cert that you can buy that has its root installed in most Web browsers by default will be fine. Usually, the only verification done is that you must be able to reply to an e-mail sent to the DNS's administrative contact, thus "proving" that you "own" that domain name.
You can use those cheap-o certificates on non-GoDaddy servers, sure, but you'll probably have to install an intermediate certificate on the server first. This is a certificate that sits between your cheap-o $30 certificate and the GoDaddy "real deal" root certificate. Web browsers visiting your site will be like "hmm, looks like this was signed with an intermediate, you got that?" which requires may require an extra trip. But then it'll request the intermediate from your server, see that it chains up to a trusted root certificate that it knows about, and there is no problem.
But if you are not allowed to install the intermediate on your server (such as in a shared hosting scenario), then you are out of luck. This is why most people say that GoDaddy certs can't be used on non-GoDaddy servers. Not true, but true enough for many scenarios.
(At work we use a Comodo certificate for our online store, and a cheapo $30 GoDaddy cert to secure the internal connection to the database.)
Edited in italics to reflect erickson's insightful clarifications below. Learn something new every day!
