Both are grammatically correct, but there is a slight difference in meaning.
As Yuri mentioned, there's a slew of Stack Exchange posts on the topic (e.g., 1, 2, 3) but even English teachers get confused about this. This BBC post for "Ask About English" is rather straightforward. The upshot is:
Cathy tried to tell her husband and son to leave the house, but they didn't wake up.
Both verbs are in the simple past tense. Cathy tried to do something. Her son and husband did not do something. One thing happened, then another didn't happen, and it was all sometime before now. That's it.
Cathy tried to tell her husband and son to leave the house, but they wouldn't wake up.
Would is both the past tense and the subjunctive mood of will, for which the OED offers 52 primary senses. There's a horde of meanings, some more odd and obscure than others.
The original and primary sense is to want, desire, choose. Here, that would mean that she attempted to speak to them but they opted or chose not to wake, in the manner of a kid half-consciously banging a cheap alarm clock when it goes off at 7 am on Saturday morning. This doesn't seem to fit here: they aren't seemingly conscious enough to be making any decisions.
By extension, it came to mean to habitually want, desire, choose and therefore to have the habit of. Here, that would mean that she attempted to speak to them but, as was their habit, they remained sleeping. That seems closer, but it involves knowledge of their usual habits or a judgment about their personalities... neither of which seem to show up.
By extension, it came to mean to be able to, to be capable of, to have the potential to and—without going into the rest of the word's senses—that's what's going on here. She tried to wake them but not only did they not wake up: they were incapable of being awoken. Since she and we would only know that if attempts were made to awaken them, @Qvalador is right that it implies there was some effort involved in a way that didn't doesn't.