Asymptotic complexity of Combination sum problem vs Coin change problem
The solution to the coin counting problem uses memoization as a way to decrease the exponential nature into just polynomial time.

The trick here, is that the amount, is always decreasing and can be computed from all of the smaller leftover amounts. this ensures that we need to compute up to amount values, that every next value is easy to compute from the ones before it.

Now for the main question: you are right in that getting the subsets themselves is what makes the difference - now the amount of 'values' you need to compute is the number you can calculate from the coin problem (for example, calculating there are 150 possible ways to distribute the coins, means you need to actually find all of the 150 solutions).
this number is not as small as amount and can be exponentially big in it

Shortly: calculating the number of solutions can be easily bounded by poly time, but calculating every solution means doing at least O(number of solutions) operations

I hope that helped! im new here :p