Neither "destiny" nor "fate" implies anything positive or negative -- they only imply predetermination, inevitability. "Destiny" is from the same Latin root as "destination," related to the root meaning "stand" (as in to stay, to go unchanged) -- to destine something is to make it stand firmly and completely, to determine it unambiguously. "Fate" means a predetermined fortune or lot in life or the power that predetermines your fortune; the word literally means "that which is spoken," i.e. by an oracle or by the gods.
If there's a difference in the meanings of the words, it's that "fate" implies something that is dictated by higher forces, something out of your hands; whereas "destiny" can mean that, but can also perhaps convey the sense that your future is predetermined by your own nature or capabilities, for instance, someone being "destined for greatness" because of their talent, personality, or ambition, or a plan being "destined to fail" because it's ill-conceived from the start. Destiny is just where your path is taking you; fate is more specifically where you're pushed or led to by forces beyond your control.
Of course, there is "fatal," which simply means "of fate," and that's part of a tendency to use "fate" to mean specifically a bad fate, as in "seal one's fate." (which is the same sort of shorthand as using "luck" to mean good luck even though there's also a bad kind of luck). But "fatal" doesn't just mean "deadly;" it carries a connotation of inevitability -- something fatal is something that will inevitably kill you, that will fix that as your fate.
So what's predestined in Destiny? Well, there's the time loop that created the Borg. There's the Caeliar's insistence that all those events had to be allowed to stand, that they were the predetermined course of things. It could also refer to the Borg's belief that they were destined to assimilate the entire galaxy. But I suspect it applies more to Hernandez and her journey, a destiny in the less fatalistic and more individualistic sense -- surviving against all odds and ascending to a great triumph.