In French,

Concurrence

Is used as a subject or object as la concurrence,

Dans le milieu des mathématiques, la concurrence entre chercheurs est féroce
In Mathematics, there is a fierce competition between researchers

Here, it means indeed competition.

Ce traitre est passé à la concurrence
This traitor now works for one of our competitors

Here, it is the set of all the competitors ; primarily of a company, but by analogy it could also be a rival sports team, band…

It is also used in the prepositional phrase en concurrence, to mean that some entities are competitors, again, it is primarily for companies

Airbus et Boeing sont en concurrence
Airbus and Boeing are competitors

Compétition

Has almost the same uses as concurrence, but is used primarily for non-business-related activities,

​Га́рри Каспа́ров et Анато́лий Ка́рпов étaient en compétition pour le titre de champion du monde d’échecs​
Га́рри Каспа́ров and Анато́лий Ка́рпов were competitors for the world chess championship

Dans le championnat du monde d’escrime, la compétition est féroce
The competition in the world fencing championship is fierce

But it can’t be used to mean the set of all competitors in a non-business-related challenge, for that, one has to use la concurrence. Instead la compétition conveniently refers to the competition itself

Une compétition de jujitsu
A jujutsu tournament

Etymology

Actually, according to the TLFi, compétition comes from the English competition, which comes from the Latin con+peto. Peto means, amongst many other meanings, “trying to reach”. Ultimately, it seems to trace back to PIE *peth₂- (“to fall; fly”), which gave feather (and then pen) in English.

On the other hand, concurrence, according to the wiktionary, is the Medieval Latin concurentia, which comes from the Classical Latin concurrens, present participle of concurro, for which I found many meanings in many dictionaries that seldom agree with each other

• According to the Wiktionaire (fr.wiktionary), concurro has the only meaning of the French concourir (compete)
• According to the wiktionary, it means “running together”, “flock”… with no mention of competition
• According to latin-dictionary.net, it means “charge”, “engage into battle”, “rally” or “run together”
• According to Perseus Latin Word Study Tool, it is “to run together, assemble, flock together”
• According to Félix Gaffiot’s dictionary (and the next page) it has the three meanings of “running together toward a point”, “meeting” and “colliding”

I suppose that the meaning of competing comes from running challenges: several individuals running together toward a single goal, and that by extension it has come to mean the competition rather the act of running.

Concurrence doesn’t always imply that there is a competition or a rivalry (but it doesn’t imply either that there is none). It means that there are others (and the term also is used to designate them as a whole) who are trying to achieve the same goal and with whom you may collaborate or just be indifferent about their success or failure.

Now there is a second meaning where there is a rivalry. Looking at the historical notes of the TLFi one see that the second meaning is newer than the first. And it doesn’t really surprise me that the meaning of a word evolves so that it becomes more (or less in other cases) specialized.

Le mot concurrence est utilisé quand plusieurs personnes ou organisations ont la même activité ou des activités similaires, chacun cherchant à être meilleur que les autres.

Exemple : Dans un service, il y a le plus souvent de la concurrence entre les employés, mais il n´y a pas de récompense précise à la fin.

Le mot compétition est utilisé quand au bout d´une activité précise il y a une récompense à la fin.

Exemple : Dans une course de 100m, les athlètes sont en compétition pour le titre final.

• Concurrence has the ultimate aim to eliminate the other.
• Competition does not aim for that. It has the aim to improve beyond the standards of the other, and hence it also contributes indirectly to the improvement of the other.