The two terms are often used as synonyms, but they come from different arguments. Strong AI is John Searle's 1980 label for the position that a sufficiently well-programmed computer would have a mind, understanding included. AGI is a capability claim: a system that can do any intellectual task a human can.
Why the difference matters
A system could be AGI by every behavioural measure and still be, in Searle's terms, all syntax and no semantics. Equally, a system could be credited with understanding in some functional sense while failing to match human breadth. The terms answer different questions: what can it do, versus what is it like to be it.
The Chinese Room
Searle's thought experiment is the origin of the split. A person in a room follows rules to produce perfect Chinese replies without understanding Chinese. If the room passes every test, does it understand? Strong AI says the system does. AGI is silent on the question.
How to use the words
Use AGI when you mean capability and breadth. Use strong AI when you mean the philosophical thesis that running the right program produces a mind. Most headlines blur them, and most disagreements follow from that blur.