"Want to" is much stronger. If you're willing to do something, it means you don't mind doing it. For example, "I want to be rich, and I'm willing to work hard to become so." "I don't really want to clean the bathroom, but I'm willing to do it if nobody else will." I hope that helps.

"want to" shows an active desire to do something "to be willing" means you agree to do it, but it's not a strong desire

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.