The difference between preterite and imperfect has to do with the manner of describing a situation, either from the point of view of the end result or the process. As Guifa says in his answer, poder specifically means mere capability when in the imperfect; it means achievement when in the preterite.
No pudieron terminar means they were (after some time) unable to finish, and the speaker is explaining the end result: no achievement. No podían terminar means they were (presumably) trying to finish, and failing, and the speaker is focusing on this process.
Imagine we're talking about a dessert.
If I was so full I just couldn't finish my dessert, I would describe the final situation by saying: No pude terminar el postre. So the dessert is or was presumably still there partly untouched, and that's the end of the story.
But if was interrupted e.g. by someone calling me on the phone, then someone knocking on my door, etc., so that in the end I just couldn't finish eating because I had to go, I would later describe the ongoing situation at that time by saying: No podía terminar el postre. The story might continue at that point. If not, the hearer would probably eventually ask: Y al final, ¿pudiste terminar el postre?, because the imperfect leaves room for doubt.
Truly, in the example you gave, you could have used the imperfect, but the context strongly suggests otherwise; if you used the imperfect, the story would sound unfinished. So they were at a given time unable to finish, and it rained a lot, and? Did they finally manage to finish? Did they wait until after midnight and resume playing?
Examples with affirmative poder, let's see:
- Al llegar a la cima pudieron ver claramente el pueblo en el fondo del valle. "When they came to the top, they could clearly see the town at the bottom of the valley." Meaning: they climbed up to the top, and became (more or less suddenly) able to see something they had up till then unable to see. They achieved something (the seeing) and that was it.
- Mientras subían desde el valle podían ver de a ratos el río. = "As they climbed up from the valley they could see the river from time to time." Meaning: as they climbed, they were on-and-off able to see the river. We don't know if, at the end of the climb, they were still able to see it.
- Podían sentarse donde quisieran. = "They could sit wherever they wanted." They were able and permitted to sit whenever they wanted; this is a fact about their ability to choose any place. (Note the subjunctive quisieran.)
- Pudieron sentarse donde querían. = "They could (were allowed to) sit wherever they wanted." They were able and permitted to sit anywhere, and did; this is fact about their final choice of seats. (Note the indicative querían.)