game and play? speak and write different?
Most of the time, "game" is a noun. "Play" is both a verb and a noun.
GAME: As a noun, it is anything widely considered fun, you could play a game of cards, a game of chess, a football game. You can also joke around and try to liven up a boring activity, then you are "trying to make a game of it".
As a verb, it is used almost exclusively to describe what serious gamers (usually video gamers) do. Never tell a gamer that he/she "plays" video games --nerd rage can get ugly. Gamers game; the don't play. Gamers are much too serious about their hobby to call it "play".
Also as a verb, it is used in place of "play" when the situation is negative or serious. If you find a dishonest, yet legal, way to get money from the government, you would be "gaming the system".
PLAY: [There are too many uses to elaborate here, so I've listed the most common. I'd suggest you visit dictionary.com for more uses.] As a noun, it is a performance written or performed for an audience, as in a theater. "Play" is also anything done in fun or in jest, "Don't worry. I said it merely in play." It is also the action or conduct in a game, "My favorite rugby team are the All Blacks. I really admire their gentlemanly style of play." It can also mean a specific act, especially during a sporting event, "Come on, team. Let's get ready for the next play!" It also refers to puns -- "word play".
As a verb, it means: to engage in a game, to give a theatrical performance, to act the part of a character in real life ("play the fool"), to employ in a game ("he played his highest card"), to perform on a musical instrument, to use as if in a game by manipulation ("play two people against one another"), etc.
Now, for your second question... Speak vs. write? This is much simpler. Speaking is oral, writing is in print. Assuming you don't have synesthesia, if you can see it, it's writing. If you hear it, it's speaking. The one exception is sign language. Deaf and hard of hearing people use a visual language to communicate. This takes the place of speaking, and is thus considered speaking.
GAME is a noun. PLAY is a verb but also a noun
Game - (an activity or sport) e.g ball games, basketball game, playing games. (wild animals/birds) Game is also the name for wild animals or birds that people hunt for sport
Play - This verb is generally used to describe actions or activities to do with fun e.g playing with a ball, playing cards, playing chess. You can also use play in these examples : play a trick, play the piano/guitar, to play in film, to play a part.
Play is also used as a noun, most commonly used in theatre - it's a piece of writing performed in a theatre or on television or on the radio. Shakespeare wrote many plays.