Which is the more correct: soccer game or soccer match? Are they both equally valid?
Note that in some sports, notably tennis, there is a division in which "match" occupies a higher place. For instance in tennis, the players engage in a match, which consists of multiple sets, which consist of games.
Game and match mean the same thing when this hierarchy doesn't apply: when a single game is considered to be a match.
Which one is used depends on the region: both the region where the sport originates, and the region from which the speaker comes from who describes it as either a game or match. One-game-match sports from Britain tend to attract the "match" terminology.
Something of a generalization, but if you're calling it soccer then you're most likely in the USA, or a place whose English is influenced by American English (since the USA is by far the most populous English-speaking region in which "football" means something else). So team sporting events are generally called "games", soccer included. But MLS does use "match" at least sometimes, so I suspect US soccer fans do likewise sometimes.
If you're speaking British English then formally it's pretty much always a "match", even if for some reason you're calling it "soccer" rather than "football". Football itself is "a game" ("the game of football", "the beautiful game"), but a particular fixture is "a match" ("did you see the match?", "I'm going to the match").
The word "game" is also used when talking about the event in a less formal way ("he had a great game") and for variety. Quoting a BBC news report that I'm watching that just reminded me of this question: "A capacity of 79 thousand people will be in there, watching the game, but around the world would you believe something like a billion people all over the planet will watch this match on television".
By the way, it's not exactly incorrect to call it "soccer" in British English (AFAIK the term was coined in England meaning "asSOCiation football", as distinct from rugby football). But it's unnecessary to specify, and "football" is preferred by almost everyone. Calling it "soccer" suggests that you refer to something else as just "football", which means you're either some kind of foreigner or really devoted to rugby.
I don't know what happens in Australia, where "soccer" would often be used to distinguish from Aussie rules football. If I had to guess, they would call it a "soccer match" there.
Game or match? This depends on what you mean. I hope this adds a bit on your understanding of the two words...
When you say game, it could mean the game itself, not necessarily played, being played or about to be played. It is simply the sports with the rules/regulations/etc.
When you say match, it means a game that was played, being played or about to be played by competing teams.
- I love playing the soccer game in my PS Vita. - The game itself
I love the soccer match between the X and the Y in my PS Vita. - Here the game was played
The soccer game was played by the X and the Y. - Here the game was played so now you could say it is a match.
- The soccer match between the two teams was great! - Again, this means the game was played