When using "et" or "mais", the two parts of the proposition have an equivalent level of importance. With this use of "que" the first part is a context for the next part. What follows "que" carries the factual statement of the sentence, which is to be read with the context in mind.
The sentence starts by reminding about the context:
La ville ne s'est pas encore remise de la tragédie qui l'a frappée.
And then describes a new event, that is reinforced by the context:
L'ombre d'une nouvelle guerre plane déjà sur elle.
You are supposed to empathize with the inhabitants of these city who just went to a tragedy and are already preparing for an other one.
You can notice that in this particular sentence the "new event" is not really an event, just a situation. This mistake from the writer is very common, particularly in newspapers.
The general meaning would be similar with "et":
La ville n’est pas encore remise de la tragédie qui l’a frappée et l’ombre d’une nouvelle guerre plane déjà sur elle.
But the meaning that the past tragedy makes the present menace even more terrible is lost.
With "mais" the sentence would oppose two things with a similar tone: it wouldn't work at all.