Dedicated IP's are desirable if you wish to access your website directly (bypassing DNS resolution): "http://IP-address", and/or if you prefer a separate stream of DNS activity to your website. Accessing your website in this fashion is not possible with a shared IP, as multiple websites are linked to it.
In general, there is no difference in the performance of dedicated and shared IP addresses, with a few theoretical exceptions. An example would be for email hosting - a shared IP means there is traffic for many mail servers resolving to the same IP address via DNS. So, if one of the hosted email accounts on a shared IP sends or receives a highly excessive number of emails, it could impact the availability or performance of the other requests to that address. Minimum bandwidth is guaranteed for each of our email hosting clients, so that your service would not be disrupted in this example. The expected activity of shared IP accounts is established through understanding the business needs of a given client, and by bandwidth monitoring if it is not immediately obvious.
Previously, a dedicated IP was required to get a SSL certificate for your site. The rise and adoption of Server Name Indication (SNI) technology resolved this issue. There is practically no security difference for shared vs. dedicated IPs, and in 99% of cases, a shared IP will satisfy all of your needs.