As I posted this, I saw that Donald had offered a similar reaction. But since I'd already written this up...
It's a very interesting topic you've raised, and one on which philosophers have written a great deal. My view fall into the blunt, even philistine category, but I'll point to other views as well.
Let's begin with your final question: is this worth getting worked up about? My answer is that it's not. What's at stake is whether some highly abstract, theoretical, and hard-to-settle metaphysical claims are true. Even if they are, life will go on as usual. You'll still experience things, remember things, look forward to things, make plans, carry them out, and in general live a human life. If there's some sense in which there isn't a single "person" that lives this life, the most psychologically healthy response is probably a shrug.
You ask whether our consciousness continues while we sleep, or whether, on the other hand, it stops and restarts. One way to read that (probably not the best way) is whether there's some mental "thing" that exists continuously throughout our lives. If that's the question, the best answer is that there's no good reason to think so. But unless we work ourselves into a philosophical lather, there's also no good reason to think that there has to be such a thing for you to be the same person who went to sleep last night. On another, probably better reading, the question is whether there's always some sort of conscious experience going on inside us (I'm treating dreams, for example, as conscious experience for this purpose) even when we're deeply asleep. Though I don't know for sure what the answer is, my impression is that there isn't. But it's not clear why I should care. Why think that for me to persist, there must be an unbroken stream of experience?
People are complicated. We have bodies with brains. The brains, it seems, are what lets us have thoughts, feelings and experiences. The goings-on in our bodies (brains included) fit together in ways that are deeply fascinating. We've learned a lot about how the more straightforwardly biological aspect work, and also how the minds that our brains give rise to work. Some of the things we learn lead us rightly to think of ourselves differently. For example: we know that we're neither as rational nor as psychologically unified as we might have thought we are. That's interesting and important. It also may be better for some purposes to think of ourselves as a complicated set of processes than as "things," but even if it is and even if we do, life goes on.
The Buddha taught a doctrine of "no self," which seemed to mean that there is no persisting inner object worth being called The Self. That's quite plausibly true. And Buddhism's view that we get out of whack when we get attached to the passing show of experience, wanting to control it and bend it to our will, is in my experience a wise teaching. Matters of living wisely aside, many philosophers have offered views of personal identity that have a lot in common with the Buddha's; the 18th century Scottish philosopher David Hume ad contemporary philosopher Derek Parfit are two notable examples. But very few of our human concerns call for getting the right take on tricky matters of metaphysics. When you think about it, that's a very good thing.