Difference between weak and strong AI
The usual distinction is that strong AI, also known as artificial general intelligence, refers to systems that can perform arbitrary tasks requiring intellect, whereas a weak AI is only able to perform specific intellectual tasks.  So, for example, a computer chess program or a system that was able to compose interesting music would be a weak AI, since it only does one thing.

Passing the Turing test would indicate something close a strong AI. Although one could argue that it only requires performing a specific intellectual task (i.e., conversation), that conversation could range over just about anything that humans can intellectualize about.  Indeed, the Wikipedia page I linked above mentions the ability to pass the Turing test as a component of an operational definition of strong AI.