The simple answer is simply to describe the awareness of the fight. I've been doing martial arts for three years now, so I know that your mentality shifts a lot depending on what style you are using. However, there is a common theme of looking for a weakness in your opponent and exploiting it. While I have never been in a plain 'fist-fight', I would assume the mentality there is more just about trying to hit your opponent as hard as you can, and not caring or knowing where or how to hit them.
The more detailed answer is below:
Before you address martial arts in fiction, you want to avoid the combat trap I (and apparently many others) have fallen into. If this doesn't apply to you, great. Skip this section.
The combat trap is essentially where an author will lapse into relating all the moves of the fight. You see this a lot where the fight is supposed to be fancy, with a lot of 'moves' involved, but it applies to any fight.
He ducked the blow, slipping under Bob's outstretched fist. He quickly threw out his own fist in a fast jab, sending it into Bob's ribs with a satisfying crunch.
Bob grunted, doubling up, but lashed out with a right hook. He slipped backwards, barely avoiding the blow. Unfortunately, the sudden movement upset his balance, and he staggered backwards. Bob, seeing his opportunity, leapt after him.
He was off-balance; any block he tried would be weak, so he simply dropped to the ground. Bob landed on top of him, but he was already squirming to the side. Bob tried wrapping his arms around his neck, but he wormed his own arms under Bob's, breaking the choke.
The problem here is that repetitive telling - while okay in short bursts - is dry, often hard to understand (if you can't describe the moves well enough), and eventually boring if it continues the length of most fights. On top of that, the fight by itself rarely adds anything to the story, unless the moves themselves are central to the plot. The trick to fix this is to focus on the character doing the fighting, and not the fight itself. This makes the fight more personal because you can see the protagonist's thoughts as he fights (which prevents it from being dry and boring), and also can do something for the story, as the character can realize things and make decisions during a fight he otherwise wouldn't. If a particular move is hard to explain, you can simply leave it up to the reader's imagination, and focus on the thoughts of the fighter instead:
He ducked the blow easily, a little surprised. What was Bob thinking? He was a professional boxer; Bob had no chance. He didn't want to hurt Bob; a quick reminder would do. From under Bob's guard, he sent his fist forward in a quick jab, feeling it connect with Bob's ribs with a crunch. Enough to hurt, but not seriously injure.
Bob grunted, doubling up, but quickly lashed out with a right hook. Caught off guard, he stumbled backwards, barely avoiding the blow. He was off balance, he knew it. He had to get his feet under him, but now Bob was lunging forwards, perhaps sensing his advantage.
There was nothing for it. He couldn't block anything while he was still off-balance. He dropped to the ground, and felt Bob land on him a second later. Got to get out! he thought. He knew if Bob got his full weight on him, he would have a hard time escaping. He squirmed to the side. Bob wrapped an arm around him, trying to choke him, but he writhed his arm under Bob's, twisting at the elbow to break the choke.
You can see how the character is much more involved in the fight, while all the moves are still related.
The same trick applies to martial arts. Focus on the character rather than the action. There is an added level of difficulty in describing martial arts though. That's because a lot of the moves are genuinely hard to describe to someone who has never seen them.
In a fraction of a second, he lined up his shoulder with his opponent. He could see the unprotected area on the chest where he would strike. His left foot forwards, he brought his back foot to the front and leapt, propelling himself high into the air. From there he had all the time in the world. His opponent's guard went up, leaving his chest wide open. All he had to do was turn so his other side faced his opponent, wait for gravity to start pulling him down, and then shoot his right leg out in a straight line, sailing cleanly into his opponent's chest. Dead center.
If you know of a way to plainly describe complex moves, by all means use it (and let me know too). But usually, you have to let the reader imagine the majority of the move, and only describe the finishing product. That is, after all, what is important.
He stepped forward and leapt into the air, spinning around as his opponent's guard went up, opening his chest to attack. He waited a split second while gravity took over once more, and then, when he was level, shot his foot out, catching his opponent in the chest. Dead center.
You can see how it's a lot easier to understand. Add in focusing on the character rather than the actions, and you have it.
Your question deals specifically with the artistic elements of martial arts and how to distinguish it from regular combat. As I said in my opening paragraph, a lot of that comes down to awareness.
Again, disclaimer: I haven't been in an actual fist-fight. However, I would guess that the discipline to break down your opponent's guard and figure out where and how to strike is not there. Maybe if you're a professional boxer. But for someone with no training, I doubt it.
Martial arts is all about figuring out how to get past your opponent's guard. If you can't get past it, then you need to make him move it to create an opening. For Tae Kwon-Do, this can mean kicking off to the side so he moves to block it, then kicking fast from the other side while he is unprotected. In Jiu-Jitsu, this can mean pretending to struggle to choke from one side, when in reality your other arm is getting into position.
There's also a lot of discipline involved. In Jiu-Jitsu, you can wipe yourself out fast if you struggle against your opponent. Knowing exactly what you need to do to keep him from choking you or breaking your arm, and doing nothing else can help you relax, as well as allow you to assess the situation and figure out how to distract your opponent, or get out of the hold. For example, you can spend a lot of time trying to pull at the arm that is choking you. In reality, you just need to get your arm under it and create a gap for your chin. With your chin in there, it's impossible to be choked. You can then relax, and start getting out.
Describing the discipline, awareness, and the breakdown of the fight will go a long ways towards differentiating martial arts from regular fist-fights. Contrary to popular belief, martial arts is not some sort of 'dance'. It is combat. The only difference is that it has focus and direction. Describe those, and describe the emotions and thoughts of your characters, and you'll have a martial arts scene readers will want to read.
Best of luck in your endeavors!