If footwork is critical to your style (and I have yet to see one where it isn't) then footing, the ability to trust your connection to the ground and to efficiently move your body when you press the earth is pretty damn important. Most martial artists practice with good footing as a given. Not always- some take care to play outdoors and on uneven terrain. Some who frequently practice outdoors in all weather learn about slick wet grass and sometimes gripping mud. Knee deep snow. Broken rock. And good footing isn't always good footing. Almost every judoka I know has broken or dislocated his little toes on soft mats.
Finished concrete with a light layer of dust is slippery. Very slippery if you are wearing good boots.
Finished concrete is also a good reflector. There is an element of hunting or ambush to a lot of fights. Practice in paying attention to shadows and reflections can give you the edge no matter which side of the dynamic you play.
Falling- You learn to fall on stuff that is much softer than concrete. I'm cool with that. With diligent practice ukemi work just fine on concrete. I've taken a full flip on asphalt at about 30 mph without a scratch and a full power throw on concrete with a 260 pound man landing on top without a bruise. The skills translate, but mistakes or poor skill has a much higher price when the surface is harder.
Impact tool and pain compliance- the earth is an impact weapon. IMO, it's usually stupid to hit people in the head with your fist. I find it stupid and inefficient to do so if there is a substance much harder than your fist right next to the threat's head. Wall or floor, driving a head into it probably works better than hitting with your hand. For one thing, you can get the body weight of both people into the strike. For another you can sometimes entrain the threat's flinch reaction into the force. Someone flinching into a door jamb demonstrates amazing short power.
Rough concrete also hurts quite a lot. There is a huge difference between kneeling on an opponent's jaw on a mat and kneeling on the same jaw when it is backed by gravel over concrete.
That's all pretty basic and obvious if you think about it. Here's what I'm really getting at:
There are things that are true on mats that cease to be true on concrete. More than that- the mats imply a lot of other background that change everything.
I trained with a man, one of the best in the world at what he does, who insisted that the ground and pound was the "worst possible scenario." Probably everyone knows, but just in case- you are on the ground, on your back. The opponent is straddling you and raining down blows to your face. This 'worst possible' scenario wouldn't even make my top ten. Five or six guys kicking would make it worse. A knife would make it worse. Being face down would make it worse. Being on top could be worse if I think I'm in some kind of wrestling match and the threat has decided it's a knife fight... on and on.
The position has a lot of emotional resonance, though. It's considered one of the most dominant positions in that sport. Many people have seen fights in gradeschool end up this way. A lot of people feel helpless when they are straddled and go completely defensive, hoping it will just be over. In a training hall, it is a very bad position. Less so on concrete.
People trying to take your head off don't hit like people trying to score points. They also don't hit like people who are trying to skillfully maintain some defense and balance. Power and rage. This is one of the reasons why blocks and covers don't work quite the same against an assault (even assuming you can get them in place in time). Practiced against someone sparring, playing all the details of range and maintaining a defense and a critical distance line and focused on preventing harm to himself as much as on dealing damage, the defenses are more communication than barriers: Punch - parry- 'hmmm, that line is closed, try something else'. A full power swing will often blow right through this type of defense. A 'beat' style fencing parry probably wouldn't work all that well on an axe.
Then the concrete comes into it. When the backstop is concrete anything that misses has a price to pay. Fully committed, full power, threat can cause his own crippling injury. You just have to make the threat miss. Shifting your pinned hips is usually enough.
The environment is part of it, but people don't think (much less fight) the way that they train or spar. Because of that disparity, another layer creeps in- what you are training against is not what you may have to use it against. In some cases, if you recognize it and exploit it, that makes things easier. The ground and pound is a bad situation in the ring, but it is one of the cases (if you are ready for it) where real life gives you some advantages.