Translation:Jenny watched a movie and went to bed.
The verbs are synonyms:
уснуть is also used in figurative meanings. Заснуть is primarily used for people and animals. It would be odd to say "Город заснул" (though, such use is still found).
уснуть, I think, has a slightly positive connotation. If you think it is bad that someone fell asleep, both can be used but "заснуть" would be more popular (I saw examples of both).
currently, they enjoy about the same popularity.
IF you mean that she went asleep, you'd say «заснула». Of course, the sentence with "смотрела фильм и заснула" would still be rather a theoretical example than a real utterance because this is not a common way to say that someone fell asleep while watching a film (more like a vague cause-effect implication).
(There are going to be lots of errors here, sorry)
Ah ok. So "went to bed" doesn't really (necessarily) mean went to sleep. I could say:
She went to bed and read a book (Она легла спать и читала книгу)[probably very wrong]
But if I truly wanted to say she fell asleep while watching a movie would I say:
во время фильма она заснула
Er... «легла спать» means she went to bed while fully determined to sleep. :) The lights would probably be off, she would get under the duvet, close her eyes and try to sleep.
Yes, your wording works, though it sort of implies she was not in control (if she watched the film from her tablet or on her PC, it would be a bit weird, since these sound as if "she" were the person playing the video).
I apparently can't reply below a certain level. But geez, I got EVERYTHING wrong, didn't I! By the way, I really appreciate how you're answering here. You're not feeding me the answers, but giving me clues (and I'm screwing them up!). Of course that could just be for your own entertainment ;)
And I'm literally, what was the phrase I saw in the comments earlier? под столом? I'm laughing, thank you, under the table. OK, so. (this will hurt your sensibilities)
She went to bed and read a book:
она лежала в кровати и читала книгу
is that better?
To correct the other, is it enough to just do this?
она заснула смотреть фильм
I'm sure I'm missing something. or a lot - there's so much to learn. Sorry.
"лежать в кровати" is, essentially, to be in bed. If you want to stress that she got into bed and THEN read a book, use perfective лечь («легла в кровать / в постель»)
Заснула смотреть фильм does not work simply because "to fall asleep" is not a sort of a verb used with infinitives to state a purpose or something. In English "I fell asleep to watch a movie" is grammatically correct but does not make much sense.
When expressing sequential events in Russian, you use the perfective aspects of the verb (which is what was used here). She watched the movie and then she went to bed afterwards. You could say she was going to bed and falling asleep at the same time, but that would require the imperfective aspect of the verbs - она смотрела фильм и (одновременно) засыпала. If you used the imperfective for one but the perfective for another, it would mean the imperfective action was interrupted - She watching watching the movie and fell asleep - Она смотрела фильм а заснула. I know I am using a different verb than what was used in the exercise but that's because like you pointed out it would look weird for her to be watching a movie and lying down to sleep at the same time (though I guess it's not outside the realm of possibility if she needs an audiovisual sleeping aid >_>).