Are these basically the same as each other?

No...for example let's say f(x) = x²

And...if p and s are the same, let's just call them both "p" .

 

First scenario:

g(x) = f(x) + p

g(x) = x² + p

 

Second scenario:

g(x) = f(x + p)

g(x) = (x + p)²

g(x) = (x + p)(x + p)

g(x) = x² + 2px + p²

 

f(x) + p  ≠  f(x + p)

 

If f(x) = x, then yes they turn out the same.

But they are not always equal.