There is some difference between older, traditional forms and the contemporary sport of Wushu, and there has been a great deal of evolution of the modern sport since the mid 20th century.
Interested in the characteristics of each, and also rationale behind the evolution to the contemporary sport.
Contemporary wushu focuses on performance.
Traditional wushu focuses on self-defense theory and application (or at least it should).
So in contemporary wushu, every movement is "flashy" to improve its appearance and level of difficulty. While the techniques themselves are taken from traditional wushu, the emphasis is on getting more air, more rotations, more extension, more distance, etc. That's because more points are awarded to more difficult motions. The actual form may have no real martial content. It ceases to be a collection of techniques that are put together in a very deliberate sequence for some self-defense scenario. Instead, it's a collection of techniques to maximize competition point count.
As I hinted at, traditional wushu forms are put together in a specific sequence in order to deal with some kind of self-defense situation. There may be 1 to 3 techniques in a form that, when strung together in that order, make a single self-defense scenario. Contemporary wushu will extract one of those techniques and puts it in a new form with any number of non-related techniques. In doing so, it loses that original self-defense purpose.
What makes contemporary wushu look like contemporary wushu is just a matter of evolution over time. The art has been evolving and will continue to evolve. What looks cool and is in fashion today may change in several years. The rules and the standardization of the point system and techniques have caused it to look the way it does. The IWUF has a series of videos on its web site that every judge must see. It shows what a butterfly-twist, for example, should look like, as well as how many points to award or deduct for different ways it is performed. It's all standardized.
Whereas, there is nothing analogous for traditional wushu. Each traditional style of kung-fu will have its own criteria for what constitutes good or bad form. For example, it's considered bad form to lean forward if you're doing a traditional northern style, but it's considered good form if you're doing a traditional southern style. Within traditional styles, though, there is room for flashiness, but that's not the focus.
Traditional wushu often has with it a breakdown of the self-defense techniques within the forms as well as a separate study of sparring and chin-na. There may also be chi-kung added as well. And there may be bag work and strength training. These don't tend to be taught in contemporary wushu. Stance training is also emphasized in traditional wushu to a much higher degree, because it's thought to be the foundation of self-defense skill, which is not the focus of contemporary wushu.
That's all I really wanted to say. As for the specific ways contemporary wushu has evolved over the years, that's a much broader topic. I think that deserves its own question, and maybe someone with a much better understanding of its history can answer that.
Hope that helps.
Practitioners of traditional wushu, as a modern person would interpret this term, probably do not call what they practice traditional wushu but something more specific. Traditional wushu is basically a catch-all category for Chinese martial arts that are not contemporary wushu.
Traditional martial arts were suppressed in Communist China during the Cultural Revolution for years. It became illegal to teach martial arts, and many people were shipped off to the countryside to do manual labor. Due to this environment and its lasting effects, the best places to study traditional Chinese martial arts can counterintuitively be outside mainland China, in places where continuous practice and instruction was available.
Traditional Chinese martial arts are decentralized. Each instructor can basically graduate whomever they choose, and graduates can do the same for any of their own students. There are no form standards or rules to follow beyond your teacher's instruction. Combine this decentralization with times when transportation was difficult, literacy rates were relatively low, and video was not available, and you can get a high diversity of styles of varying quality. There is often no sport format for traditional Chinese martial arts.
Contemporary wushu is the result of a Chinese government initiative to revive martial arts and is highly centralized. There are national training programs, and you can study wushu at university.
The simplest way to understand contemporary wushu is as martially-inspired gymnastics. There is a regulation-sized floor area where competitors perform routines (forms). There are judges who score these routines with skills rated by difficulty and execution. There is an outsized emphasis on adding combinations and more degrees of rotation to jumping moves. There are different events such as long fist (changquan), southern fist (nanquan), staff, straight sword, taiji, xingyi, etc. Although the events may match the names and general appearance of traditional styles, the goal of wushu is to have good-looking forms, not to have forms that teach you to fight.
The differences are due to reforms made during and after the Cultural Revolution in China. During this period militant efforts to purge "traditional Chinese" culture to reinforce Maoism, which led to the destruction of a lot of martial arts manuals, documents, and treatises. Three million people were subject to be killed specifically because they were well known teachers of traditional Chinese culture, and this included well known martial art practitioners. That's not mentioning its anti-Buddhist purges that targeted many Buddhists (and martial practitioners).
By the 70s, interest in martial arts from movies made in Hong Kong by Ip Man and Bruce Lee rekindled interest in kung fu/Wushu practices. In the 80s, kung fu movies took off successfully in the US home video market. And new actors Jet Li and Jackie Chan also help rekindled interest in kung fu/wushu as a practice. Monks would travel the world.
Nevertheless it's a shell of its former self. Masters struggled to exploit new teaching tools and rediscover what they knew. The Chinese government used martial arts as a tourist draw, to market their teaching styles, and prospective students are being lured away by the draw of mixed martial arts (MMA), a combat sport that is exploding in popularity and requires no standard technique to implement. As a result, few fighters think of kung fu as a legitimate martial art any longer. Today, kung fu is a blend of martial technique and various isometric exercise, stretching, and elaborate gymnastics.
There are some very good answers and I want to supplement with some historical perspective.
After Communism, there was a goal of creating a national sport by standardizing Chinese martial arts in the form of Wushu. This did indeed result in oppression of traditional schools, but it's also important to note that the group who created the first generation of standardized form all came from traditional lineages. (The 67 combined Tai Chi form is a good example of integrating core techniques from most of the major styles of Tai Chi.)
The original standardized forms were still focused on martial application, as opposed to "flash".
It's also important to recognize that Chinese martial arts were never separate from performing arts, as "medicine shows", where practitioners would perform in the streets for coins, were common. Thus flashy, crowd-pleasing tricks have a long history that predates modern Wushu, and it's doubtful that serious practitioner didn't recognize the difference between these and practical techniques.
After the initial period in the 1950's, contemporary Wushu evolved further, with the goal of becoming an Olympic sport. My understanding that this is the main driver of what mattm astutely described as "martially-inspired gymnastics". And it's not that those practitioners are not strong, just that the contemporary art is not combat-optimized, but optimized for judging in routine-based competition, like ice skating and gymnastics. (Even the early standardization facilitated judging by introducing compulsory forms.)
Tai Chi is an instructive a good example in that, at a certain point, competitors were required to make their back straight, which is antithetical to the martial practice of tai chi, to facilitate judging—specifically determining that the weight is definitively on the proper foot. (This was the point when my own teacher, a top competitor in their own right stopped having students participate in competition.) High-level Hsing-yi often involves concealing where the weight is, and this obviously presents problems from a judging standpoint.
Also notable that KungFu movies goes back to at least the 1920's. The original stuntpeople came out of traditional schools, and later the Chinese studio system set up its own schools to train actors. During the Jet Li era, the Beijing Wushu team became a feeder into the Chinese action-film industry. (The original Shaolin Temple features the best competitors and teachers from that era, and led to Li becoming an international star.)
But it's also important to recognize: top contemporary wushu competitors, like gymnasts, have exceptional core strength, are incredibly fast such that it’s unlikely that they can be engaged if they choose not to engage, are exceptionally flexible and train in a range of movements which allow them of get out of locks, and, if thrown, are arguably the most likely to land on their feet, which also has application in getting out of some locks.
The exception is Sanshou/Sanda, and you do seem some Chinese Sanshou champions in MMA. But my sense is it's not as emphasized because there is not the national goal of Sanda becoming an Olympic sport.
And, again, Chinese practitioners, both performers and fighters, have an historical pathway into stuntwork and film, which is typically less punishing than prizefighting. (Jackie Chan, trained in an opera school, is famous partly for having broken most of the bones in his body, but avoided traumatic brain injury. Chan has had a significantly longer career than any modern prizefighter, and remains one of the wealthiest martial artists in history.)
