Panini is a Plural, the Sandwich is Actually a Panino
In America, a panini is a sandwich that has been grilled and pressed. Right now, you may purchase a sandwich at any sandwich store. You can prepare them at home by purchasing a panini press from Walmart. Paninis for all! It's important to note that using the word "panini" in this manner is not even grammatically correct. Multiple sandwiches are referred to by the Italian word panini.
A panino is a single sandwich of this type. Thus, panini would be two. When you go to Subway and order two paninis, you're actually trying to get two sandwiches, which is incorrect in terms of how to refer to these sandwiches.
It's important to remember that panini is Italian for "little bread roll." Thus, it refers to any sandwich that is served on the full little loaf of bread. In North America, grilling is basically the only requirement, but it doesn't even have to be done.