On the electrical distribution side, there are a few things which may need internet connectivity. Getting map tiles for displays; having an externally contracted call center integrate with the outage management; automatic vehicle location (painting crew vehicles on the displays).
Adding to the last point, something that is coming to the industry is read only access to the system from a mobile device. This requires internet or a private network across the bounds of your network.
I don't want to talk too much about company specifics, but typically modern systems will have servers dedicated to only internet related functions. They will be internally firewalled from any servers which could make changes on the network. These systems aren't cheap though, a lot of what we replace is 15-30 years old. As such, it may not be as secure as it could be.
I've mainly talked about distribution. Transmission and generation also have functionality which requires the internet, or at the least a very large private network.