This question has been asked before. I tell those who ask this question that there are differentiations and there are also commonalities. Differentiations are: 1) MMA is a sport, 2) MMA has certain rules, and 3) MMA is constricted to a consistently controlled environment. Do these restrictions automatically make a fighter less prepared in a self-defense situation? Not generally. Factors such as one’s upbringing and life experiences cannot be isolated from such fighters.
Martial arts on the other hand, from its universal and primitive point of view: 1) It is not a sport, 2) It is not confined to a particular controlled environment, and 3) It does not consist of the rules and laws predicated on the athletes’ safety as outlined by various organizations or commissions. The rules and the laws in the context of ancient combative history were forced by the environmental conditions (pressures) on humans which had led to a martial-based natural selection so to speak, founded on instinct to survive the prospect of a much bigger loss than what we would expect in modern MMA.
Also, with the exception of few systems of martial arts which are intrinsically focused on developing athletic fighters——the rest have unfortunately become a pool of ineffective philosophies and their techniques have been reduced to mere conjectures and assumptions, and are no longer based on the current collective fight experiences or full contact sparring.
The common ground or the authentic unifying vision regardless of the terms Martial Art or MMA is that neither the striking nor the grappling were invented in 1993 UFC. The principle mixture of the systems and how they influenced each other is quite ancient, going back to the antiquity and beyond. Ancient China used to have a system called Leitai which had a mixture of boxing, grappling, and other elements. In Greece, obviously there was Pankration. Similar mixture of ideas were also prevalent in ancient India, Egypt and Persia. And these were not the only people who had such ideas, if you dig deep, you would find that they were around in other geographical regions on the planet as well no matter how obscure and distant the civilization.
Whether you are a genuine martial artist or a MMA fighter, in either case you still need a solid development and understanding of the common attributes such as timing, distance control, composure, power, rhythm, base, coordination, and so on. Without these universal attributes, nothing would work anyway, regardless of what you like to call yourself.
In short, if you see yourself as a MMA practitioner because you like a mixture of things, but at the same time you tend to reflect and appreciate the history of the arts, and like to improve and advance them (not just advancing yourself)——and if you like to apply your skills in a self-defense situation, then you are already a martial artist. And if you call yourself a martial artist, and you put on the gloves and the fight gear and spar all fight ranges, then you are already a MMA practitioner.
To be complete, you need both mindsets which are two sides of the same coin.