We have a competitive-gaming, and we have a esports. These are basically the same, and should be synonymized, but the question is, what direction should they be synonymized in? Should competitive-gaming be used, or should esports be used?
I wouldn't trust Wikipedia when common sense is at stake. It should be also noted that competitive-gaming tag has 3 questions and esports has only 1. It's not a good enough number to judge which is more popular among users, though.
When you're not familiar with terminology, you're going to try and describe something with words understandable by largest audience. And I don't think most users would expect a site such as SE to predominantly use topic-specific terminology, when it's main goal is to be friendly to as many users as possible. That's why we don't see as much lingo used for programming sites as tags' primary names. Surely, each area of human activity has its own terminology, but that doesn't mean we have to go full academic on people. That wouldn't be nice.
I vote for competitive-gaming.
esports makes the most sense as the primary tag.
esports is the term used by tournaments such as DreamHack and MLG. Wikipedia also has a page on Esports, and redirects Competitive Gaming to the esports page. The top Google result for Competitive Gaming turns up the Wikipedia esports page as well.
I see these tags as entirely different subjects as noted in both answers.
esports as a genre is the video game equivalent to talking about the "Majors" (See Baseball, Basketball, American Football, Futbol, etc) as the U.S. recognizes Video Games players from other countries as "Athletes", this is where I would expect a question such as "What are the rules regarding the MLG variant of Team Deathmatch for Call of Duty: Ghosts?".
Whereas competitive-gaming is the industry of Competitive Gaming where questions such as "Which companies fund the Texas MLG Tournament?"
But this is my personal preference and even though I see the difference I also see the similarities in them.