I do both, actually. I found with very low carb, the state of ketosis works to curb your hunger--which is great! But if I went by the "only eat when I'm hungry" rule on low carb, my calories ended up being SUPER low (800-1000 calories a day, anyone?). I now count both my carbs and my calories. It's interesting to me that on a diet where I'm so satisfied all the time that I have to count to make sure I have
enough calories, as opposed to making sure I'm not eating too many.
I've done both calorie-counting and low carb independently of one another as well. Low carb works best for me personally--but I count my calories, too. I use Fit Day, log everything... It's not so bad once you're used to it

It all comes down to a personal choice, and what you can sustain. In my opinion, everyone can benefit from reducing their carbohydrate intake and upping their protein and good fats--no matter what diet plan they're following. To the degree that people moderate carbs is up to them--I like ketosis-level carb restriction, some people use Zone-levels and hit a more moderate 30%. It really depends on you.
EDIT: One thing I didn't say, and I meant to, is that there is no magic about whichever way you choose. No one diet is going to be the magic cure for obesity or health--it's still your hard work and determination that gets you to your goal eventually. Depending on your lifestyle and how your body reacts, one diet or another might make the journey
easier or
harder for you. But no diet is going to work over night, or in some mystical way.