On a website dedicated to wild and crazy ideas, this one is a humdinger. I like it!
- In the absence of a central star, the world would eventualy have a rotational period equal to the lunar orbital period. (This might be true even with a central star... but at least there's something injecting energy into the system to keep it going.) This results in the geostationary orbit discussed by @HenryTaylor in his comment (what Henry thought of as a simplification is actually what would happen). Here's the twist, if the rogue planet was "spun away" from its parent star "recently" (I'm going to ignore what "recently" means in astronomical terms), then you would still have a difference between the two periods. The longer the rogue is away from its parent, the closer to a geostationary orbit. I'm assuming t's fairly "recent" so the difference still exists. Should the time come when the moon and planet lock up their periods, you'll have daylight on one side, no light on the other, making the other basically uninhabitable (too cold).
BTW, you could have the rogue pick up the moon as debris from its travels, but that begs the question where all the happy earth-like life came from? Could the evolutionary sludge of life have existed in the cold of space for however many eons the rogue wandered the lifeless void? It really complicates the backstory. So, I'm assuming the spin-off backstory.
- I'm going to hang on to your orbital and rotational periods for as long as I can. The Earth rotates once per day, Luna orbits the Earth once every 27 days. Scaling the issue, your rogue rotates once each day and its moon orbits 36 times per day. That's an orbital period nearly 1,000X faster than our Luna. (Granted, I'm cheating for dramatic effect. Without knowing the difference in rotational period between your rogue and Earth, the calcuation is presumptive. But it is very dramatic....)
However, now that I think about it, this really isn't the primary issue. Without a point of references (big ol' sun in the sky, or at least some stars), the rotation of the planet is entirely irrelevant. If we maintain that a "day" is one period of dark and one period of light, then a "day" is two orbits of your moon at 18H per orbit or 36 hours (which, according to Men in Black can cause the entire population to experience a psychotic episode).
- For the convenience of this discussion, I'm going to assume the moon has no inclination. This means it's always right over the equator. @A.C.'s answer points out the biggest problem with this assumption: massive polar regions. However, adding an inclination to overcome this has another problem: reducing the overall lighting affect on the planet. This whole situation is just plain ugly and will likely be the cause of saying this can't work, but we'll see.
People would likely die from epilepsy (I'm kidding, but look at that graph!)
- You now have a rogue world with an equatorial band of life-giving warmth and very large polar ice caps. Your "day" is 36 hours long. Your "great day" is 648 hours long. During most of that time you're going to see pulsating light to one degree or another. Here's a chart of how much light a person would see if they stood on one point along the equator for an entire great day:
Another way of looking at this is that it's "high noon" at one longitude only once every 648 hours. This is nothing more than the product of three sine waves: planetary rotation, lunar orbital period, and illumination cycle. It does not take into account refraction through an atmosphere (which would heighten the backside illumination pulses).
Life would, as at all times, adapt or die. I have no idea how such a light source would ultimately affect anything.
But what about inclination?
Ultimately, what this does is widen the habitable band around the equator at the expense of lowering the average amount of light seen equatorially at one longitude. Your world would be colder, but there's more world (theoretically) to inhabit. I'd almost say crank up the heat and use a high inclination... but during those high pulses the world would be HOT! In the long run, I don't think an inclination does anything but harm the chances of life.
And what about the weather?
I'm not able to easily imagine this, either. You'd have very calm, honking cold weather around the very large polar caps, but you'd likely have serious wind around the habitable region with that pulsing light pushing it along like a combustion engine. If the habitable land has a high percentage of water, you'll get serious storm buildup. This wouldn't be an easy world to live on...
But I think it could be habitable, with complex flora and fauna. But it would be a miserable life compared to the eden of a planet in orbit around a decent sun. Natives would love it and think the sunlings weak wossnames, while the sunlings would never visit such a planet but for a day to "experience" it.
I forgot to explain what back-side illumination is: it's the illumination seen by that one point of longitude when the moon is illuminating the other side of the planet. This is because, starting with high-noon at our reference point, nin hours later the moon is on the other side of the planet at 50% illumination. Nine more hours and it's back over the reference with 0% illumination. Nine hours again and it's the backside with 50% illumination, etc. (This example assumes no rotation of the planet. The graph does.)
Edit: If we're trying to replace the earth, then we need "solar" radiation equal to 1,000 W/m2. The earth is 510 million Km2 for a total energy requirement of 1W/m2 * 510x1012m / 2 = 255 Terrawatts continuous power. (Yes, I didn't bother with the curvature of the Earth reducing the total absorbed power... but it won't the last simplification, either.)
The RMS value of the normalized impact of Sol on the Earth is 71% (I skipped a couple of steps to explain where the % comes from, but it belongs there). The RMS value of the curves in my graph is 25.66%. That means the moon must generate 2.77X the power delivered to the rogue planet the Sol does for Earth.
The problem is, That's outside the max 1.5X habitable zone. (Click here for appropriate newscast.)
So, our fist solution is to reduce the peak solar impact to 1.5X. That gives us an adjusted ambient power delivery of 54% Sol average.
Eureka, the habitable zone!
So, our world will be colder, but still habitable and not suffering from the snowball effect. Deserts are dry, but not hot. The temperate zone is smaller. The rogue planet happens to be the same size as the Earth... but who's counting? But we're alive and enjoying the ride.
Getting the overall energy absorbtion closer to 100% of Earth norm requires changing the duty cycle of the moon illumination cycle. I don't know if you want to go into that much detail (you certainly needn't).
The power needed to move the moon around is trivial compared to the power needed to energize a habitable planet, so you can ignore the problem of geosychronous locking.