Of course is used say that something is obvious. Is Bill Gates rich? Of course he is. Does a bear shit in the woods? Of course. If it's going to be a great party and/or you've been looking forward to it for a long time, you might say "Of course, I wouldn't miss it for anything". For sure means definitely, or without any doubt (undoubtedly). You could certainly say "for sure" as a response to that question too. "Are you going to the party?" For sure. Without any doubt. If something is obvious, then there probably isn't any doubt about it. So they usually convey a very similar, almost identical meaning. I don't think that either is better than the other in this case.

If someone asks you for permission or asks you for a favor, you can answer with 'Of course' or 'Sure.' The meaning is the same. It's a friendly way to say 'yes.' Can I borrow your cell phone? Of course! / Sure! Can you help me with my homework? Of course! / Sure! If someone offers you something (food, drink, etc.) then I think 'Sure' is a better answer. In this case, 'of course' means you really like something and it should be obvious to the speaker. Would you like a glass of white wine? Sure! (Of course is a little weird here unless everybody knows you are a huge fan of white wine.) I have two tickets to Hawaii. Do you want to come with me? Sure! / Of course! (Both OK here since it's understood that anybody would say yes to this offer.) Others have said that 'sure' sounds very American, and I think that's true. My British and Irish friends have also told me that it's very American of me to say 'sure' all the time.

None really, you can use both. Usually "of course" it's used as an answer to a question, or to state something you have said or done before. Like, for example: "Sam told me to go to McDonalds and ask for a beer. Of course I wasn't going to do it, it goes against my principles." In this case you don't use "sure" But as an answer to a question: "Would you go to the market and buy me some oranges?" you can use "sure", or "of course".

In response to a question they both mean the exact same. Can I use your toilet ? Of course ! Can I have some milk ? Sure ! Although from my observations, I have noticed that Americans tend to use 'Sure' a lot more than in Ireland.