The "cloud" versions of the software are exactly like the software that came on optical discs. It installs on your machine and runs on your machine just like the older versions used to work. The only difference is that you don't have to mess with discs and you need to connect to the internet every month or so to keep the software activated.
It's really fast to install the software too - much faster than the old optical disc way of doing things. Once it's downloaded (which doesn't take long on a decent connection), the software installs within a minute! Also, you can use your software on any machine using either Mac or Windows and then switch to a different machine at any point. It's much better this way.
Another huge benefit of the Creative Cloud (especially if you work with other people) is the Libraries feature. You can set up library and share it with your team members who are working on a certain project with assets particular to that project. Logos, color themes, stock art, etc. all in the same spot for everyone to access.
Even if you work alone, the mobile apps (like Adobe Capture) work with the Libraries really well to help you grab assets as you travel. I was on a trip in San Diego and walked into thrift shop. I saw an old quilt that had a color scheme that would fit perfectly for a project I was working on back home. So I used the Capture app to make a color theme based on it in just a couple of seconds and then, when I got back to Nashville, it was sitting there in my Adobe apps ready to be used! (I really like the Capture app.)
Don't forget Typekit too! As a CC subscriber, you get a huge amount of fonts for free. You can install them and start using them without having to exit Photoshop or After Effects or whatever you're working in.
If I had a choice, I would not go back to optical discs. Instant access to any version of the software I want as far back as (and including) CS6 on any computer I want is great.