Is a subgraph either a spanning subgraph or a full subgraph?
On your question "isn't a full subgraph actually a spanning subgraph?". The answer is no, a full subgraph doesn't need to be a spanning subgraph. By your definition, a full subgraph can have lesser number of vertices than in the original graph. However, a spanning subgraph must have exactly the same set of vertices in the original graph.

On your second question "how to calculate the number of spanning subgraphs of $G$ as a function of number of nodes in $N$." Since it is a fact that graph $G = (N,A)$ with differing $A$ will have differing number of possible spanning subgraphs, I suspect you might have misunderstood the question. For example, does the question actually only ask for an upper bound of the number of possible spanning subgraphs as function of $|N|$?