This isn't too easy to differentiate but examples will help:
A pattern is a description of something which can happen more than once.
1. You can have a pattern for a dress from which you can make lots of copies.
2. Your behaviour can follow a pattern, where you always do the same things over and over again.
3. The pattern on the wallpaper repeats every 50cms.
A mode is being in some particular state. Each mode may involve a sequence of events or instructions. The sequence of events could be called a pattern.
During the week I operate in 'Work' mode: that is I get up, have breakfast, go to work, come home, do the housework, go to bed. The pattern of behaviour is same every day when I am in this particular mode.
At weekends, I operate in 'Play' mode: sometimes I sleep all day, sometimes I go and watch football. Sometimes I go shopping.
A mode is a state of being, you can switch from one mode to another.