Telligent - Verint Cloud/SaaS

From a platform/software perspective it is the same, just managed by Verint.

We've been on Verint Cloud for around 3 years. I can't speak to the differences between self-hosted, but here are some thoughts:

We have access to edit the core code of basically every widget, automation, theme, etc. through the admin panel. Any of these can also be imported/exported through the admin panel to move them between sites (e.g. almost everything from Marketplace can be installed by an admin). There are some things that require Verint's support team to install, but these are usually simple processes and are easily solved through a support case. APIs should also allow you to fulfill most/all of your existing integrations. 

I work for a company that offers cloud and self-hosted options to our customers and I also have admin responsibilities on other Customer Experience platforms that my company uses. Verint's cloud offering allows SIGNIFICANTLY more access/flexibility/configuration than any other platform I currently or previously used. Some of the devs at my company that help out with some of our community work have been really happy with the openness of the platform. 

There is no difference software wise from on-premise to Saas.  The difference is all infrastructure, meaning in Saas Verint is managing the servers and infrastructure and the tasks that go along with that.  With that anything that requires physical server action requires a support case.  On-Premise means you are responsible for not only software upgrades, etc, but also the management and patching of the infrstructure.