Why call it discrete time signals as difference equations? There are lots of representations of discrete time signals. You represent it in block diagram or as difference equations.

We call it block diagrams as there are elementary structures in the diagrams. 

Why do we call the equation part as difference equation of a signal?
$y[n] = x[n] + x[n-1]$

There is nothing getting subtracted in the above example representation, Then why call it difference equation??
A difference equation is a recurrence relation where the current element $y[n]$ of a sequence is related to its past values $y[n-k]$, $k&gt;0$ (if we interpret the index $n$ as time index, which is not necessary). The equation you gave in your question is a special case, because there are no past terms $y[n-k]$ involved, so you could call it a zeroth order difference equation, but you're right that from a common sense point of view it is no difference equation at all.

However, what counts here is the fact that no past (or future) elements $y[n-k]$ are involved, and not that "nothing is getting subtracted". E.g., the following equation is a standard example of a difference equation:

$$y[n]=ay[n-1]+bx[n]\tag{1}$$

The subtraction takes place in the index $n$ ($y[n-1]$). But obviously you can substitute $n+1$ for $n$ yielding

$$y[n+1]=ay[n]+bx[n+1]\tag{2}$$

which is completely equivalent to $(1)$ with no subtraction taking place, but it's still a difference equation, because the difference between the indices remains unchanged.