Most of the time, I have heard them being used interchangeably, so there is no difference really. Maybe, it is just your preference to use one over the other because that's how you learned it. For example, I only use tie because that's what I'm used to saying. I am not too into sports, so I can't tell you if there is a difference in the sports usage, but in general usage, there isn't.

Cricket is different. In cricket a tied match is VERY VERY different to a drawn match. It is far more common for a match to be drawn, in test cricket anyway compared to tied. I will try to explain the difference Tied = the teams have scored the same number of runs e.g. India vs England. India score 200 and England 150 in the first innings. Second innings India score 200. If England get bowled out for 250 runs exactly then and only then is the match tied. Tie has a specific meaning here. The teams have to get bowled out having scored EXACTLY the same number of runs. A draw is different, it just means the match could not be completed before either team takes 20 wickets, or scores the amount of runs required. This is far, far more common in test cricket. In other sports, draw is the word more commonly used in the UK when teams have the same points, or number of goals etc e.g. football, rugby etc. In the USA they use tie for the same thing I think. As I said before cricket is a special case where they mean different things.

Draw has the same meaning. Use whatever you like.

I like to think that ties mean "Everybody wins" or "These two or more people win" Where draws are "Nobody wins"

A draw means there wasn't a clear winner. A tie means they were equal.