I just signed up with DreamHost VPS, and their sign-up process offered an unique IP address for an additional ~$4/mo.
I know what IP addresses are. Why would this uniqueness matter? Visitors are accessing my website via URL addresses anyway.
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The big one is that you need a unique IP address for some SSL/TLS implementations. As pointed out in the comments, no version of IE on XP can do this, which is the biggest offender.
Also, if you have an application that needs to reference an IP instead of a DNS name, you'd need it since your shared host is likely configured to ignore requests to an IP.
Blacklists (SEO, mailings etc.) are often based on IP, so having unique IP will reduce propability that you will suffer for someone's other actions.
If you don't know that you need a unique IP address, then you probably don't need one.
Some protocols and applications require a specific IP address, but for the ones most frequently used these days, host name alone is often sufficient.
Obviously, in the background, this must resolve to an IP address, but not necessarily a unique or fixed address. As long as the server understands enough to route the query appropriately, all the services could share the same pool of IP addresses without issue.
Another benefit is being able to point multiple domain names at your static IP address in order to run multiple stand-alone websites that each resolve to their own folders.
I agree with Bill. If you don't know you need it, don't get it. You can always add it later if you find you need an
ssl certificate (like for processing credit cards).
As for the comments you may hear online about security of having your site on a unique IP so that you are not lumped in with spammers, for example, almost everyone in the world is on shared servers. Most hosting companies now are good at responding to spam issues and making sure they get rid of bad clients.