A Mary-Sue will have extreme emotions and interactions with characters. They go from 0 to 100 faster than a formula one race car.
What I mean by this is a Mary-Sue (or Gary-Stu) will suddenly have a deep hatred for a character they just met without sufficient reason or time.
Of course, this can go the other way and they immediately have the hots for someone who just text them 'hi'. Quite frequently it's the other characters that automatically fall in love with the Mary-Sue. They adore them and are more likely to risk their life for this Mary-Sue character than they would be for their family.
I should also point out that a Mary-Sue is often overly confident and never doubts themselves when talking to other people. They always say exactly the right thing when flirting or when aggravating someone else.
I advise building up relationships between characters, rather than just blowing up a relationship immediately. Also, don't be afraid to make your characters stutter or get nervous when talking to people. We all get like it at some point, it's human nature.