is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...? Is there a relation between the fact that the derivative of x^2 is 2x and that the difference between 1,4,9,16, ... is 3, 5, 7, 9, ...? 
And why is the difference always 2?
I think there is a relationship, but I can't get how and why...

Yes. The derivative of $x^2$ tells you how fast the function $x^2$ increases. The second derivative of $x^2$ tells you how fast the derivative of $x^2$ increases. The second derivative is 2, meaning that the speed at which the rate of growth of $x^2$ increases is fixed. Therefore, the difference of the difference of consecutive squares is constant.
For your second question, let $d_n = (n+1)^2 - n^2 = 2n+1$. Then
$$
d_{n+1}-d_n = 2(n+1)+1 - (2n+1) = 2n+2+1-2n-1 = 2.
$$
