On my Trek roadbike, I have Continental Duraskins and they've been the best tires I've every had on a road bike. They're fast rolling and very sturdy, almost flatproof.
On my Allant, I've already replaced the rear tire and need to replace the front too as well as I saw today there's big gash in it. For it, I'm going to buy a
Schwalbe. I had Schwalbes on my Mt bike and they were indestructible and were able to carry a pretty heavy load. The stock tires on the Allant are ok for around town riding but they aren't as good at carrying a lot of weight.
You might notice some difference but if the Hardcase's selling point is puncture resistance, then you probably won't notice much of a performance increase. The more puncture resistant a tire is, generally the beefier it is.
Personally, for around town, I prefer a beefier tire. They can take potholes and curbs and debris a lot better than true race tires.