Dissatisfactory would be in regards to an opinion or emotional response.
I.e., I was dissatisfied with the soup, but the chicken was delicious.
Here, the soup itself may be delicious for one person but not another - implying the person found the soup dissatisfactory due to his/her taste but the chicken was to their liking.
On the other hand, unsatisfactory would be in regards to the object or item not meeting the standards expected. To use the same example:
The soup was unsatisfactory so we asked for it to be taken off the bill.
I would interpret this as the soup was simply not to an expected standard (similar to your game example). I would say though, at the end of the day both of these examples (and your example) are opinion based so I would always write it as:
I was dissatisfied with the game - especially the plot - but I still enjoyed the level design and the atmosphere.
I hope this has helped highlight that in the first two examples, the words certainly are interchangeable, as shown; however, the meaning and underlying tone of the sentences has changed.
I was unsatisfied with the game - especially the plot...
Now I have demonstrated the words are interchangeable by making the words linked to my opinion.