The SSP concept started in the late 90's. The goal was to make storage similar to electricity. You plugged into the wall and there was your storage. In general, cloud storage follows this same model. Both are based on a pay-as you-go model. As far as similarities go, though, that's about it.
What's different? The first big difference that comes to mind for me is the infrastructure that the CSPs are using. Most, if not all, of the SSPs used very expensive, high-end, name brand hardware to host their customers' data. The idea in those days was to impress the end user with the storage brand name behind the SSP. We see almost the exact opposite today, CSPs are more often than not leveraging their name first and the hardware second. Although the quality of the hardware is important, greater emphasis is placed on the policies and procedures that these companies employ.
The next difference is availability and cost of bandwidth. Bandwidth is everywhere now and acceptable speeds are very affordable. Those connections tend to be a lot more reliable. I can't remember the last time our office internet connection has gone down, but it has been at least a month or two, and when it did, it was only for a few minutes. Clearly better availability can be had than our standard vanilla office connection. Also, if you project bandwidth out, we are not far from near ubiquitous bandwidth all at very high speeds, and in a data center we are there now.
The hardware model itself is also different. SSPs tried to use tier one, monolithic solutions that were really designed for single-server, or at most, single-organization use. The SSPs became software developers in their own right, trying to address the shortcomings of a monolithic storage system and to try to ease their own management burden on them.