The biggest difference: The second derivative of quadratic growth (the change in the rate of change) is constant. While any derivative of exponential growth stays exponential.
For quadratic growth, think naive sorting like bubble sort. For exponential growth, think bogosort.
Technically: If you use Big-O notation, then indeed O(x^2) \subset O(a^x). But also O(x) \subset O(x^2) \subset O(a^x).
