When Jesus calls God the Hebrew word for Dad or Daddy, Abba, he is revealing that he has a personal relationship with God, which goes beyond the fact that He is God, but also a caring father-figure. This is important because it shows that Jesus knew that he had a unique relationship with God. Only He knew Him well enough to reveal Him to others, and link us to God. When Jesus calls God dad, he is getting on a personal level with God, something none of us could do without Christ's help. Because of this, we don't have to lower ourselves as much with God, because since Jesus showed us we can get on a personal level with Him, it raises us up to be able to personally speak with God as a friend, not just a king. It is also important because it shows the personal level of Jesus when he isn't worried about the sanctity of humanity, and that He is also a normal person who just wants to talk to His Dad about personal things.
All of this ties together because it shows how we aren't just impersonal worshippers in a kingdom, but loved children who can have a personal relationship with their king, and give us confidence on how we are situated in God's eyes. As far as how the word Dad is used in today's world, children pretty much accept it as everyday language, as a way to personally identify with your father. Nowadays, to say father is considered to be impersonal and ultra-polite, because it is considered to show that you don't really know your father that personally, and you can't identify with him. To say dad is considered to show that you are comfortable with him, and that you have a personal enough relationship that you can identify with him in ways that you can't with anyone else.