Demo like you build your frontpage

I have just started to come to the realization that my front page was all about the problem / solution of the customer (or at least that was the objective) but when it came time to demo, I showed them a list of my features.

Introduce the same problems you display on your landing page and solve them in the demo. They don't necessarily care about the features, and within Saas new features are coming out all the time.