The term "Cyber Warfare" is largely nonsense. There just isn't enough there to make a prolonged exchange of hostilities likely, not on the order of magnitude that you could call a "war".
However, as we have seen already, there is quite a bit of critical infrastructure reachable (directly or indirectly) through the Internet. If an actual war would break out between any non-3rd-world countries, it is quite possible that the initial strikes would include cyber attacks to disable as much of that as possible.
At the current level of tech actually deployed (i.e. ignoring fancy future dreams about everything-IoT, etc.) that would mostly be it. There is a very simple countermeasure against cyberattacks: Go offline. Most critical infrastructure can run without Internet connection. Of all the various companies in this area that I've done security work for in the past decade or so, none would stop working without Internet. Sure, it would be a big hassle, but power stations would run, fuel would still run, traffic systems would run, trains would run, the airport would still be open. All at reduced capacity, all with major operational trouble, but they would run.
The most affected part would be supply chains, which rely strongly on data exchange today, but there are still enough old people in the business to switch back to older methods (phone, fax, and if you have to, couriers).
You could damage a countries economy as a lot of consumer stuff and B2C trade relies on the Internet today. But we also have a vast parallel infrastructure of supermarkets, shopping centers and such, so the damage would be manageable.
That is not to say that there would be no damage. Entire companies would go out of business without Internet, and the impact on daily life would be quite severe, especially regarding communication which has moved so much to the Internet that most of us don't even have the phone numbers of a lot of our friends, only their various online contact details (e-mail, FB, Twitter, etc.). Is WhatsApp still using phone numbers as usernames? That might save your social network. But for anyone outside the immediate circle, including any business that you're not a very frequent customer to, most of us today look up the phone number online if we have to call them, and it's been a long time since I saw a phone book anywhere.
But in the context of an actual war, the cyber aspect would be a negliegable factor, and when you worry about nukes falling, Amazon and Google going out of business would not be frontpage news.
That said, why do we have "cyber commands"? First, for publicity and to attract more nerds to the military. The military understands smart people. Second, because all of this is changing and when you think about the future, not the quarterly-report future of the business world, but the 10-years, 20-years type of future, then certainly the Internet as well as IT security in general will only become more important, so start being ready today. Third, you want to protect your own infrastructure against such attacks. Even if you don't believe (like me) that any cyber attack would be more the equivalent of sabotage than that of warfare, it is still something worth protecting against. If you look at the budgets of those "cyber commands" and compare them to the cost of, say, a new aircraft carrier or a couple fighter jets, you see the real priority of "cyber warfare" quite clearly.