Differences between sports and karate-do In considering this question one must first provide some definitions. A challenge is that those definitions themselves frame the question so tightly that one could be accused of arguing by definition - a traditional fallacy in argumentation . Let's start with those definitions to explain what we might call the intuition of the traditional karateka about what those differences are, and then return to try to correct the structure of the argument to avoid fallacy.
We define karate-do as a physical activity in which the participants learn a particular approach to empty-handed combat. We define sport as a physical activity in which, over a defined period, there is a contest between participants resulting in a score. A score has a winner and a loser, although some sports allow exceptions such as a tie, draw, disqualification etc. The objective of a sport is to win a competition, while the objective of karate-do is self-improvement.
There are however some obvious problems even as we begin with these definitions. Many sports are practiced within the context of self-improvement as a primary goal. I ran two marathons. I had no hope of winning the marathon, placing, or even winning in some non-unitary sub-category such as runners from Napa County (I might have won in the category of runners from my street, assuming I was the only entrant in that category). Most casual observers would call marathon running a sport, and yet "running" as a physical activity would be more closely grouped with karate-do as an activity for self-improvement - improving health, mental discipline, the ability to push through physical pain etc.
A further problem is evident when we consider the obvious sport aspect of karate. Traditional karate as practiced in the JKA and ISKF explicitly sanctions and includes sport. Nakayama  included these sport aspects in his development of the system and they remain with us today. Competitions are held regularly at local, regional, national and international levels in which karateka compete in kata and kumite. There are scores and winners and losers. One might say that, like marathon running, for many of us, competition is just another means to the end of self-improvement. For those of us who lack great physical talent, victory in karate competition is unlikely, no matter how hard the training. So, competition is just another exercise - an opportunity to place oneself under a certain kind of stress in order to learn how to overcome that stress and others like it. But for the talented and hard-working individual, winning is certainly an important aspect of competition.
Further complicating this issue is the fact that sports have long explicitly included personal development as a core value. The word "sportsmanship" is taken to mean a certain honor of playing within the rules of the sport and treating the opponent with respect inside and outside of the competition itself. In many sports, especially the neighborhood "pickup games" played by adults, the primary objectives are exercise and camaraderie, rather than winning. Or we might say that while winning a game is the immediate, proximate goal, the objective in context is much larger than that.
The sport aspect of karate serves many purposes beyond the competition. In fact, it is not uncommon for some teaching karate-do as a pure art to become disconnected from a larger karate community, with the result of practicing at a lower level than those who compete, interact, and learn from others.
One might also attempt to differentiate karate-do as a "pure" art in which there are no rules. In kata we practice groin and eye strikes that cannot be allowed in competition. Competitions without rules do exist outside the bounds of modern law and gladiatorial competitions were not uncommon in the history of many cultures and countries. Even in kata however there are bounds on techniques. While we practice a "pure" notion of killing with one blow, it is evident that is an ideal that has tenuous connection with reality. Violence between evenly matched participants is more likely to end with an uneven application of force , such as use of a weapon, or the "ground and pound" knockout common in Ultimate Fighting . Even karate-do embodies rules in the application of force that expresses the sort of sportsmanship required in competition. Martial art forms such as Aikido  tightly constrain techniques so that there is not even the notion of an attack.
Now we can return to attempt to correct the faulty premise of this argument - that of arguing by definition. Definition is used to collect a family of related phenomena into a group that can act as a summary for the purpose of efficient communication. So, we must look at exemplars in the range of sport to martial arts, and the superclasses of each to see if this grouping is completely objective. We might label sports as X and karate-do as Y in order to limit the implicit associations people may have with those terms, in hopes of making the argument more explicit and objective. What things fall within X and Y? Into a bin called "X" we might place soccer, ping-pong, boxing, and fencing. Into a bin called "Y" we place karate-do, aikido, marksmanship, running, and weightlifting. Into X we add sport karate, judo, ultimate fighting, and modern taekwondo. At this point, it should become clear that we are really discussing aspects of related practices. It would be easier to create a common superclass Z to contain all these notions. Individuals often cross-train in sport and martial-art. Within sports are "pure" art aspects, and within martial art, sports competitions are used to further training.
In fact, Wittgenstien might council us that we are really just playing a "language game" . What can we really determine by describing the different between "sport" and "karate-do" other than our definitions of two words? Like karate itself, there is a paradox in that by searching for a definition, we arrive at an understanding of the issues. The goal of the exercise is larger than the immediate objective. In seeking to vanquish an opponent in combat, we learn more about ourselves. In seeking to define the practice of karate, we hope to have come to a deeper understanding of our own practice.
 Wittgenstein, Ludwig (1953/2002). Philosophical Investigations. ISBN 0-631-23127-7.
 Nakayama, Best Karate, Kondansha pub.
 Grossman, Dave, On Killing: The Psychological Cost of Learning to Kill in War and Society, Back Bay Books, 1996.
 Jonathan Snowden, Total MMA: Inside Ultimate Fighting (Paperback), ECW press, 2008.
 Eckart Köhne (Editor), Cornelia Ewigleben (Editor), Gladiators and Caesars: The Power of Spectacle in Ancient Rome (Paperback), University of California Press, 2000.
 by Kisshomaru Ueshiba, The Spirit of Aikido, Kodansha International, 1988.