Hi Vadym. Here is the difference below:
'want to' is a desire. 'I want to play a game.'
'to be willing to do something' is more like an ability. You are able to do something but you might not necessarily want to do it. 'I am willing to play a game.'
Think of it as this. If your friend asks you what you want to do today, what could you say as a response? Maybe you want to go to the movies or maybe you want to do nothing. It's light, it's fun and a desire (something you like to do). Now, think of that same friend asking 'Are you willing to help me move to a new apartment?' This might not be such a fun thing always. If you are a helpful friend, you might answer that you are or are not willing 'Yes, I am willing to help you'. In English, usually the more common things to say is 'Yes, I can help you.' as a willingness and an ability is synonymous.