Background Knowledge
Think of an interface as a contract. In a contract between two people, it defines what is capable, what is expected from the parties involved. In programming, it works the same way. The interface defines what to expect, what must exist for you to conform to that interface. Therefore, since it only defines what to expect, it itself, doesn't provide the implementation, the "code under the covers" so to speak, does.
A property behaves like a field, but allows you to intercept when someone assigns a value to it or reads the value. You can also deny reading or writing to it, your choice when you define the property. Interfaces work with properties instead of fields because of this. Since the "contract" is just defining what property should be there (name and type), and if it should allow a read or write capabilities, it leaves it up to the implementer to provide this.
Take for example the IEnumerator interface from the .NET framework. This interface was designed to allow iteration over a collection of objects. The purpose is not to change items, or randomly access them, it's just for getting object A and moving to the next, and the next, and the next, as many times as needed. Many collection type classes implement this: Queue, ArrayList, SortedList, Stack, etc. All these types of objects store many objects and now they all share the common "contract": the ability to iterate one-by-one over them.
However, you can see that the IEnumerator interface has a MoveNext() method declared. Why? This is because the items may not be served in the same manner. For example, people will generally access the ArrayList from the first item to the last. But a Stack was designed opposite, for people to access the last object down to the first.
Questions Answered
With all this knowledge, the limitation of declaring a variable as the interface type as opposed to the class type that implemented the interface is that you only get access to what the interface (the contract) says should be there. The benefit though is that you can assign to this variable any class type that implements the interface.
structcan implement an interface just the same as aclasscan.