Some Suggestions
So first off, I am obviously not someone who knows a lot about you or your friend, nor do I know the specifics of your situation, so please, take everything I say with a grain of salt, as my advice will be very general.
Now, without further ado, my two cents on the matter: Honestly ask yourself, what is the argument over? Does this disagreement go against your moral beliefs? If so, be serious with your friend and explain why (note: moral beliefs are something that you feel very strongly about, and it can be very hard to have friends who share different or opposing moral values). If this is not a moral difference issue, is the argument something you could change your own views on? The reason I ask, no matter how much pondering you do, you can’t change someone who doesn’t want to change, so always start with yourself.
Also, it is good to keep in mind when disagreeing with a friend, listening and talking equally is extremely important. One person cannot expect the other to listen if they aren’t doing it themselves. Check yourself, are you listening to why your friend thinks you are being immature? Do you comprehend how they want you to ‘grow up’? Next, check your friend, (and be real with yourself) are they listening to you? Are they comprehending your side of the argument as much as you? Both participants need to listen to and comprehend the others words in order for both to come to a standing agreement. In the case of unwillingness to listen, reevaluate how important the friendship is, and decide how much time/energy you are willing to spend trying to (KINDLY) explain your point of view to your friend.
To sum all up, decide how important the friendship is, decide what you are willing to change and ask the same of your friend. Try to reach a compromise that solves the issue while preventing future ones by listening to one another. And do please remember, this is just my opinion.
Know it’s not the Game
First thing's first: Clarify your goal. Decide what you ideally want to happen. For me, understanding is usually my goal, and that makes it easier to figure out what's wrong together. Other times, my goal is clarify why something is the way it is- whether it's feelings or actions or rules. Some things aren't up for discussion, and making those hard lines clear can be really important. Whatever it is, decide what you want to accomplish and what progress would mean.
Second, if you think you're friend is annoyed over a game, you're thinking in the wrong place. If it's truly over 'small things' it wouldn't be so hard to confront or talk about.
Ask yourself questions. Are they right? Are they being mean, unreasonable, rude? Are they doing it on purpose? Are they starting something or reacting to something? What about it is hurting or bothering me? And most importantly: where is it coming from? A big part of solving problems with other people is understanding what you're doing and feeling, so it's easier to tell people and talk about. The more prepared you come to the confrontation, the easier communicating and understanding each other will be.
Sometimes when it's about 'small things' it's a bunch of small stuff they dismissed and let slide. When you ask, make it clear you're listening. You're asking because you want to know something you don't know. You're not asking rhetorical questions (at least you shouldn't be ._.) and you're not trying to make them say something they don't mean. Be clear that your intentions are to resolve whatever conflict is there.
Appreciate what they tell you, whether you wanted to hear it or not. Always thank them for what they're willing to share. What you do with that information is all up to you. Communicate as gently as you can, and ask yourself how you can reach that goal you set at the beginning with the information you have now.
Take your time. Fixing and changing things always takes time. You're learning. Your friendships should always be worth the trouble
Good luck
Advise
First,why is she so annoyed with you over the game? Is it because you are spending time on the game when she is over or because you are spending alot of time on the game?
If she is upset because you play while she is there- it may be that she feels like she is visiting you and not getting the attention she feels she deserves and so she is upset about it- feeling neglected. If she feels like you are spending too much time on the game- she could be concerned (as a friend should be) IF you are spending a considerable amount of time on it. Put yourself in her shoes- hear what she is saying and WHY it bothers her. Then ask yourself if her concerns are legitimate. Reverse the situation and ask yourself how would you feel if the situation was reversed.
Now...with that being said... if she is making fun of you. This is different. If you enjoy something and she does not- That is ok and as a friend she should understand that. It is ok to be friends with someone who has different interests- however, if she is putting you down by calling you immature just because she doesn't like the game. She is actually the one who is being immature because she is making YOU feel bad just because you are into something different- and that is NOT cool. I am not sure what the situation is... but instead of arguing - ask her- why does it bother you that I play this game? If she says it is just because she doesn't like the game- tell her that you and she may not always like the same things, but that you can still be friends and that you do not like it when she says you are immature. Try getting to the root of whats bothering you both... tell each other how it makes each of you feel and work through it- if she is a true MATURE friend- she will do this and maybe things will get better,