"Stoppage" is a noun, whereas "stopping" is a verb; there are cases where they cannot be interchanged.
For example:
The leaves are stopping up the drain.
The leaves are stoppage up the drain.
He walked in while the laughter was stopping.
He walking in while the laughter was stoppage.
There are cases where they can be interchanged and change the sentence meaning.
For example:
- The food stoppage had massive consequences.
- The food stopping had massive consequences.
Both of those sentences make grammatical sense, but (1) suggests food blocking something, such as an airway, and (2) suggests the food ceasing to move, such as a cut-off of supply.
Lastly, there are cases where they can be interchanged and not change the meaning of the sentence.
For example:
Premature stopping of the procedure will cause complications.
Premature stoppage of the procedure will cause complications.
Here they both (virtually) convey the same meaning. This is, I believe, because the italicized words can effectively act as a single noun. If you want to get really nit-picky, you could differentiate and say the second sentence conveys that the state of being stopped will cause complications, while the first conveys that the the act of stopping will cause complications, but the difference is hardly distinguishable and can even reverse depending on the dictionary used.
The fish trap exists because of the fish. Once you've gotten the fish you can forget the trap. The rabbit snare exists because of the rabbit. Once you've gotten the rabbit, you can forget the snare. Words exist because of meaning. Once you've gotten the meaning, you can forget the words. Where can I find a man who has forgotten words so I can talk with him?
~Chuang Tzu