On the GMAT, problems about factors are invariably constrained to POSITIVE INTEGERS.
The problem above should read as follows:

What is the greatest common factor of POSITIVE INTEGERS x and y?

(1) x and y are both divisible by 4
(2) x - y = 4

Statement 1: x and y are both divisible by 4
Options for x and y:
4, 8, 12, 16, 20, 24, 28...
If x=4 and y=4, then the GCF of x and y is 4.
If x=8 and y=8, then the GCF of x and y is 8.
Since the GCF can be different values, INSUFFICIENT.

Statement 2: x-y = 4
If x=5 and y=1, then the GCF of x and y is 1.
If x=6 and y=2, then the GCF of x and y is 2.
Since the GCF can be different values, INSUFFICIENT.

Statements combined:
Statement 1 yields the following options for x and y:
4, 8, 12, 16, 20, 24, 28...
Statement 2 indicates that the difference between x and y is 4.
Implication:
x and y must be equal to two consecutive values in the list above.
If we select any pair of consecutive values -- 4 and 8, 8 and 12, 12 and 16, 16 and 20 -- the GCF in every case is 4.
SUFFICIENT.

The correct answer is C.

One worthwhile additional note here: the GCF of two positive integers x and y is itself a factor of the difference between x and y. (Algebraically, we'd say GCF(x,y) is a factor of (x - y).)

So S2 tells us _something_: the GCF of x and y is either 1, 2, or 4, since the GCF of x and y must be a factor of (x - y), or 4.

Combining the two statements, then, we know that 4 is the GCF, since it's already a common factor of the two numbers (i.e. 4 is a factor of x and 4 is a factor of y) and it's the greatest of the three potential common factors (1, 2, and 4).