I believe the two styles generally refer to uses of energy - hard or internal styles (karate/kung fu) utilize your energy for defense. Soft or external styles (aikido/hapkido) utilize your opponent's energy against them.

I've also heard of distinctions between striking linear styles to circular non-striking redirecting styles.

Most teaching lines are becoming increasingly hybridized so the traditional lines are becoming increasingly blurred or non-existent.

Hard: 1. Training method which uses force vs force as a strategy and training method. Examples include Shotokan Karate. 2. A training method placing value a simple plan of attack, a small set of tools drilled to mastery and the use of muscle tension, including the recruitment of additional muscles beyond what are needed to perform a task.

See also: Hardening. Transforming a part of the practitioner’s body into a durable weapon capable of smashing bones through a program of sustained microtrauma. An example would be to pound one’s fist on a log, telephone pole (etc) creating microfractures in your striking surface (such as hands, shins, forearms). Your body compensates by adding calcium deposits that will lead to increased bone density and mass in the effected area. Note: This produces early onset rheumatoid arthritis when practiced on joints, such as those found in your hand.

Soft: Training method which uses fighting strategies and/or training methods not based on force vs force. Redirection, yielding, avoiding and opponent’s force is emphasized. A soft style practitioner might spend time doing drills at a moderate speed to recognize and systematically shorten the response time to a tactile stimulus.

What hard/soft doesn’t indicate: How hard you work, how much or how little you think, how well you fight, what kind of shape you are in.

My own opinion, after 40 years in the martial arts, is that the hard/soft distinction is neither useful nor enlightening. Good fighters have elements that are both and sometimes neither. Consider the old Thai adage that translates to, “Be hard like diamond and soft like silk,” which is a poetic way or saying, “It’s complicated.” They harden their shins by kicking heavy bags many times (like >500 times per day). Hard style, right? Yet their footwork is beautifully relaxed. Their hitting mechanics are relaxed so they can hit with speed and power, and they learn to tighten up for the instant in which contact is made. And in the inside clinch game in which they seek to use their knees to break your ribs and collapse your lung, relaxation is highly valued, just like in judo. In the clinch, a guy who is tense up top is easy to sweep. A guy who is tense on the bottom is immobile and slow. And a guy who is tense all over is a target that will soon tire himself out.

Western boxers, aren’t really either hard or soft either. In fact, if you asked at a lot of boxing gyms they would laugh at you and tell you, “Just shut up and put your @#$% on.” They specifically avoid damaging their hands by wrapping them, so soft style, right? Not really. Western boxers employ different styles, some based on evasion and mobility (such as Mohammed Ali) while others seek to work their way inside and put pressure on their opponent with a series of powerful body shots delivered at close range (Roberto Duran, Marvin Haggler, Mike Tyson, etc.) Then again you have some fighters who do both (Pacciao) and can switch strategies to frustrate and confound an opponent (Sugar Ray Leonard).

Judo might be thought of as a soft style, but in Judo the hard/soft thing changes as people age. As a young competitor you start out using a ton of power and explosive force because that’s what young people excel at. As you age, however, you can’t take the falls as well because your bones aren’t as flexible. In addition, by the time you’re in your 30s and 40s you have your technique down so you can have what they call the ‘lever’, which is the ability to throw an opponent with minimal force. By the time you’re in your 50s you have to relearn your game because you are no longer so quick or so strong, but if you stay in the game that long you will have your technique and sensitivity (contact reflexes) down cold. As you get into your 60s and 70s your greatest opponent will be age itself. You practice because without it you will quickly grow old. But you have so much experience at that age that you can make everything look easy as if it is magic. And that’s actually the big secret. There is no magic. Only work, and perseverance and discipline.

A third reason, sounding somewhat flippant but not intended that way - it's useful when having a discussion with the layperson/buddy you're trying to get to start training with you/friend or family member that says, "You still doing that [insert style] (South American Ground Karate, in my case)? Is that anything like the [insert another style] that my niece/nephew/someone else they know does?"