What's the difference between $f \cdot g$ and $f(g(x))$? For example if
$f(x) = x + 2$
and
$g(x) = 4x - 1$
Then what would be the difference in $f \cdot g$ and $f(g(x))$?

The notation $f \cdot g$ means that for every $x$ the function is
$$ (f \cdot g)(x) = f(x) \cdot g(x) $$
which  is pointwise multiplication.
On the other hand $f \circ g$ is the composition of functions,
$$ (f \circ g)(x) = f(g(x)) \ . $$
For your examples:
$$ f(x) \cdot g(x) = (x+2) \cdot (4x-1) = 4x^2 + 8x - x - 2 = 4x^2 + 7x -2 $$
while
$$ (f \circ g)(x) = f(g(x)) = f(4x-1) = (4x-1)+2 = 4x + 1 \ . $$
