TL;DR
Gamification is the same thing as competition?
Definitely these concepts are not the same thing. Competition is an element that can be used in a gamified process, but it is not the only interesting or necessary element.
Gamification
One of the first studies of gamification in a scientific way was conducted by Deterding et al. in 2011 (the term came into fashion around 2010). For these authors, the definition of gamification is:
Gamification is the use of elements of game design in contexts that
are not games.
Free translation of the English original: Gamification is the use of game design Elements in non-game contexts.
This means that we can use techniques and tools traditionally used to build games in the design of any product (software or not) whose intention primary not entertainment. And why would anyone do that? Because, unlike that piece of code you have executed and it obeys, humans need to be motivated to act in some way - and the fun is a great intrinsic motivator (by intrinsic understand that it does not depend on obvious external benefits like money, food, promise of future earnings, etc).
Okay, but what about the competition?
Turns out this is a complex issue. The very definition of gamification has a very nebulous boundary, mainly because defining what is game is also difficult (if you are interested, read about the feud between Narratologists and Ludologists). I like the definition of Deterding et al. mainly when it is considered along with the work of Fullerton:
The image above is based on the two works cited. Deterding et al. ’s idea is that there are two axes that define the different types of products/tasks that use game design techniques. The horizontal axis, called "All-part", indicates whether the product/task is considered as a whole (i.e., as something unique or monologue) or in parts (i.e., composed of smaller things or activities). The vertical axis, called "Play-Play", indicates the user’s intention when using the product or performing the task.
Fullerton’s contribution to the detailing of the vertical axis demonstrates that the level of interaction helps further in this classification: the interaction does not just involve touching and moving, but also cognitive effort needed to recognize a goal and plan the actions necessary to achieve it. Thus, the more interaction is needed, the closer a product/task is to a game; the less interaction, the closer it is to a mere narrative (a book, a film or a play, for example).
Of course this definition is debatable, but it seems useful to me to understand that Gamification is about using these design concepts to intimately build tasks composed of several parts, but that have not only a goal as a condition of victory (important from the point of view of the user). Another clear thing in this definition is that a gamified process is quite distinct from a game for entertainment, and also from a "serious game": a game (a "Whole" product) that is used for a purpose other than mere entertainment (as, for example, when Wii Motion are used to help in physical therapy).
Competition can be part of any of these types of products, even in toys/games. Even without having a goal or win condition, a joke can still involve competition in other aspects, such as having the cutest toy (see any interaction with toys between the characters Keys and Quico on TV) or be the most able to detail the narrative or fantasy involved in the play.
Anyway, adding medals and scoring lists is only interesting if the user perceives this as something valuable to him, and - mainly - if he can compare his score with other users to experience emotions related to the competition (as was well placed by the accepted answer).
But then what’s beyond the Competition?
The complexity of the subject does not stop in the definition of what is game. It is also difficult to define what is fun and what produces it. Competition is certainly an important element in fun because it involves challenges (opportunities for action that allow a difficult but feasible goal to be achieved), but it is not the only element, nor is it always necessary. This character is one of the easiest to measure in a game. For example, in a racing game I could measure the number of times the player has hit the car or been delayed, and use this "utilitarian" measure to try to find out if the player is being able to compete or not. If I were to conclude that he’s doing extremely badly in the competition, could I infer that he’s not having fun? That question has been asked by many people, and the answer seems to be nay. After all, the player may be hitting the car on purpose, taking advantage of the ride and dazzled by the graphic effects of the beats, so that if asked maybe he said he was having a lot of fun!
Finally, there are many other aspects that are important beyond the challenge (related to the competition). There are hundreds of studies, but Malone, for example, it states that the important aspects are Challenge (clear action opportunities and objectives), Curiosity (new and different sensory information, but not completely chaotic) and Fantasy (mental images that help to understand the real or fictional world). The one MDA model (one of the most known references in the area) has already evolved this into other aspects, including the social aspect, exploration (of the world and the self), and even the mere effort to spend time (which the authors called "submission").
But, you want to have a well summarized (and fantastic) view of what is really "at stake" (Pun intended) beyond the competition? Then gives a studied in another model called The 4 Keys 2 Fun, of Nicole Lazzaro. Next I reproduce the image of Nicole’s model (the rights are all from the author, I reproduce here at fair use only with the intention of disseminating the author’s knowledge and work, which is freely accessible on the website referenced above). It is worth a lot to also access the author’s blog, read her works and, mainly, watch your talk at TED Talks.
As you will notice, this model indicates that competition (which involves the quest to win challenges, whether alone - against oneself - or in a group - against others) is only part of the fun. There is also the social aspect (interacting with other people, creating affective bonds, and spending time with friends), the curiosity aspect (allowing the exploration of fantasies and the satisfaction of the imagination) and the purpose aspect (thinking, feeling and behaving in a way that makes a difference in the world and in other people’s lives).
In your question you say:
"This proves the direct relationship of gamification to competition as
form of engagement"
I’d say that proves that competition has its function. In addition to motivating certain types of people to compete, Stackoverflow votes also have a character stigmatic by making content with more votes (positive or negative) attract more and more attention and consequently tend to gain more votes. However, a website or a website that should be gamified should also seek other aspects for people’s motivation, exploring their emotions also with social ties, with curious visual and sound elements, and so on.
It’s like something to overcome something you already have. It’s not really a competition, it’s a challenge. Interesting.
– vinibrsl
@vnbrs in the course of games, my coordinator calls gamification motivational design. I just remembered the zombie’s run, an app like this fitness that you run and, if it gets slower than a limit, you hear the horde approaching
– Jefferson Quesado
it is possible to gamify an application using rewards that keep the user motivated but without necessarily having a competition with other active users, although the competition is an interesting artifact within gamification itself.
– rodrigorf
@rodrigorf exactly! There are people who prefer to call motivational design instead of gamification
– Jefferson Quesado
Badges is a pseudo-competitive system, which when compared becomes competitive
– HudsonPH
@Hudsonph in the social perspective of "I have, you don’t have", visible badgers generate competition. Just like the simple visible scoring scheme. Pure competitive scheme even would be leaderboards.
– Jefferson Quesado