What cognitive qualities distinguish humans from other animals?

Two qualities that are central to our "humanness" are language and self-awareness. While many animals communicate, and years of training can give chimps the language level of a two-year old, no other animal uses language with the same sophistication and creativity that humans do. Also, our self-awareness means that we can think about our thinking.

This means we can assess its effectiveness, learn from mistakes, and share information with others about our mental states. We can also think about others' thinking, allowing for a higher degree of strategic behavior.

Which is more influential in communication: non-verbal or verbal language?

It depends on what your goals are in the interaction. If you are primarily interested in gaining information -- finding out when the next bus leaves, for example -- verbal behavior will be much more influential than nonverbal behavior. However if you are trying to determine whether someone is telling the truth, or forming an impression of what someone is like, nonverbal behavior is often more influential.