To find the intersection points, we set the two equations equal to each other:
\[3-x^2+x^3 = 1+x^2+x^3.\]
Simplifying, we get $2x^2-2=0,$ so $x^2=1$ and $x=\pm 1.$
Plugging these values back into either equation, we find that the $y$-coordinates are $y=3$ and $y=1.$
The maximum difference between these $y$-coordinates is $3-1=\boxed{2}.$
The answer is: 2
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
The graphs of $y=3-x^2+x^3$ and $y=1+x^2+x^3$ intersect in multiple points.  Find the maximum difference between the $y$-coordinates of these intersection points.

### Answer:
