"Chúng tôi không ngủ."
Translation:We do not sleep.
Yes there is a difference between the two. In Vietnamese, you would add the verb được, which means 'can' as in 'to be able'.
Chúng tôi không ngủ được.
We cannot sleep. / We are not able to sleep.
Actually, cannot = không thể. To say "We cannot sleep." in Vietnamese, you need to say Chúng tôi không thể ngủ. If you want it to be more natural, you can add the word được at the end of the sentence like willhnguyen said (but note that this word is not a verb) and sometimes omit the word thể. So the sentence can be changed into Chúng tôi không (thể) ngủ được.