Hmm, we're kind of hitting most of the answer but not all of it (in my opinion that is!).
Its probably better to ask this the other way round - why not use a road pedal on a mountain bike...
On a road bike you're generally going to keep your foot in the pedal, you're only going to take it out to stop so
therefore you want it to be as simple and as light as possible.
On a MTB you're liable to have your foot on and off more often, you're also likely to want to be able to stomp your foot onto a MTB pedal and have it stay there so it needs to have a larger platform area.
You could also make an argument about MTB shoes being softer but the designs tend to be driven by competition
and so that's probably more coincidence than anything else.
If we're talking about using toeclips then clearly you'll need bigger pedals and clips for MTB boots than road shoes.
And weight does matter - it may only be a few 10s of grams but a few here and a few there will add up and all other things being equal less weight will make a difference.
I use clipless pedals - which I recommend -
and I have one pair of shoes that I use on all my bikes,
some of the pedals are fairly minimal for road, some clearly MTB, some hybrid but more MTB than not and all work with the same SPD cleat.
Paradoxically my current setup has double sided pedals on the "road" bike and single sided on the MTB type pedals on my shopping bike...
So why road pedals?
Assuming clipless and further assuming that you will only wear cycling shoes then you don't need the "complexity" (no difference in my pedals) or the weight of a full platform -
so why have it?
But the real difference will be in the shoes you chose to use.
(As I final thought I actually have two pairs of shoes - one waterproof - and a pair of sandals
and they all have the same cleat so that I can use whichever is appropriate to the weather on whichever bike I happen to be out on.)
bicycles.stackexchange.com/users/69/Murph
The main difference between road and mountain shoes are the stiffness.
On road bikes, the shoes are much stiffer allowing for a more efficient transfer of energy from you -> crank -> tire -> road.
When you are in a race and every tenth of a second counts, the more efficient you are the better.
On a mountain bike, you give up some efficiency, so you don't sprain your ankle every time you plant your foot.
Flexibility is better than stiffness.
bicycles.stackexchange.com/users/32/M. Converse
You'll find many differences between MTB pedals and road pedals, just like every other component on the bike.
Here are some of the key differences: Weight.
Most road pedals are made of lightweight alloys, carbon, or plastic.
Road cyclists are more inclined to reduce weight of components than MTB cyclists.
Durability.
Needless to say, there's going to be much more abuse to a MTB pedal than a road pedal.
2-sidedness. Most road pedals can only be clipped-in on one side, whereas most MTB pedals have interfaces on either side of the pedal.
As far as I'm concerned, you should ride with whatever makes you most comfortable.
Most people will use MTB pedals because they are easier to clip into (due to the interface on either side).
I don't think the weight savings is terribly important unless you are doing some serious climbing or racing.
bicycles.stackexchange.com/users/68/Russ Warren
For most road-bikers, the difference is weight.
Modern road pedals are often much lighter weight than the typical mountain bike pedal.
bicycles.stackexchange.com/users/23/Dustin Ingram
The only real difference between road bike clipless pedals and mountain bike clipless pedals is weight and aerodynamics.
When you consider the fact that a bicycle and rider are going to weigh over 100 pounds (unless you're a midget on some insane carbon-fiber time trials bike) a few dozen grams doesn't make a bit of difference.
bicycles.stackexchange.com/users/20/kc-bike
Meh, I would say if it's comfortable for you
and you don't need to conform to specific rules for competition or anything, keep doing what you're doing.
I've been riding road bikes with mountain-biking shoes for years and don't feel like I'm Doing It Wrong.
bicycles.stackexchange.com/users/33/Scott
I am a big guy, so I ride with Look pedals on my road bike, because they have the largest cleat size, and the largest area to distribute my weight to the pedal.
Shoes vs Pedal style are two different issues.
bicycles.stackexchange.com/users/19/geoffc
I always maintained the major feature difference b/w road and MTB pedals as being single vs multi release shoe cleat.
Road pedals typically only release when you turn the heel of your foot out - hence single release cleat.
MTB pedals are multi-release (i.e. will release if you turn your foot in or out past ~45 degrees).
bicycles.stackexchange.com/users/11/kkeilman
Dirt and mud are much more common on a MTB, but not a real concern in most road riding.
The tolerances for MTB pedals are much higher than are needed for road riding, where the pedals won't be as dirty.
Also, I find myself engaging my pedals with a MTB in much more "odd" positions compared to my road bike, which is virtually always on the flat, with one foot on the ground.
On a MTB, I might be stopped on a trail pointing any which way.
All of those things are perfectly fine on a road bike, but unnecessary.
bicycles.stackexchange.com/users/389/LanceH
Researching on this myself, the main reasons I've found is that more surface (road) is better for power transmission and more comfortable on longer rides, which you would also want on MTB but road cleat can be inutilized with mud or just natural soil not tarmac, so you could have problems stepping with road shoes on a MTB track terrain then trying to reclip with stuff inside cleat.
There seem to be also range of motion (lateral?)
issues, better on MTB pedals but not so needed in road.
bicycles.stackexchange.com/users/18463/gaurwraith
I've been riding road bikes for a couple of years, but with mountain bike pedals, I've been thinking about switching over to a road bike pedal, but what's the reason? Is the performance improved or is it more about conform?
Why is a proper road bike pedal preferred for road bikes?
bicycles.stackexchange.com
<road-bike><pedals><parts>
bicycles.stackexchange.com/questions/74
bicycles.stackexchange.com/users/29/Johan Dahlin
On a MTB you're liable to have your foot on and off more often, you're also likely to want to be able to stomp your foot onto a MTB pedal and have it stay there so it needs to have a larger platform area. Most road pedals can only be clipped-in on one side, whereas most MTB pedals have interfaces on either side of the pedal. Another major feature difference b/w road and MTB pedals as being single vs multi release shoe cleat.  One of the main arguments for mountain bike pedals is the larger platform area. This allows toeclips to be fastened, even on both pedal sides. Mountain bike pedals also have a multi release shoe cleat. However, you should consider that road pedals are significantly lighter, which could lead to better performance. Regardless, it would be a good idea to take the terrain and road conditions into account, as they might make road pedals inappropriate for mountain biking.
One of the main arguments for mountain bike pedals is the larger platform area. This allows toeclips to be fastened, even on both pedal sides. Mountain bike pedals have a multi release shoe cleat.  You should consider that road pedals are significantly lighter, which could lead to better performance. More to the point, road bikes tend to have stiffer pedals, prioritizing efficiency over flexibility. Road bikes tend to have stiffer pedals, prioritizing efficiency over flexibility. It would be a good idea to take the terrain and road conditions into account, as they might make road pedals inappropriate for mountain biking.