I know it seems like it should be cheaper for Verizon to accept electronic payments, but I once worked for the company that ran their web application servers. Verizon has outsourced that job. The convenience fee likely goes to the outsourced company.
My water bill incurs a $2.95 "convenience fee" as well, and that goes to Western Union. They run the website and transfer the money. For that, they get the $2.95 per customer per payment.
The company is happy because they don't have to process the payments, run the app servers, or pay for the service. The provider is happy because they get all this money for $0 marginal cost. The customers get the shaft and don't complain loudly enough.