I assume that you are asking about connectivity in the sense of the internet rather than in terms of ships and airplanes and such.
Global connectivity’s major advantage is that it allows rapid communication of all sorts to and from practically any spot in the world. This makes trade much easier. It makes it easier for businesses in one country to deal with firms in others. This increases worldwide efficiency.
The major downside to global internet connectivity is the danger of cybercrime or cyberwarfare. The internet makes it much easier for people in one country to try to scam those in other countries. It allows countries to try to attack one another’s infrastructure using viruses and other malicious software.