There is no real difference, but most people tend to relate "contact with" to physical actions/availability. (Examples: "The man's came into contact with that persons face.", "Stay in contact with your children. Who knows when they will need your help?)
"Connect with" tends to be used to define a relationship as personal and deep. However, it has lately been more frequently used in place of when it comes to advertising social media and the internet. (Examples: "You connect with my emotions so well", "Connect with your friends and family using *Insert Brand Name Of Social Media Here* ")
They are fairly interchangeable, so do not stress about your use of these words too much.