Mark Foden usefully distinguishes the two ideas as the difference between a clock and a cat.
Complicated things — like clocks — are discrete mechanisms. They may have many parts. With care and attention someone could understand exactly how the complicated thing will behave. Such an expert could disassemble and rebuilt a complicated thing, and it would function the same as before. The interaction of the parts is precise, regular, and predictable.
Complex things — like cats — are whole systems. The interaction is unpredictable. You can’t be certain how a complex thing will behave. An expert can’t dissect and rebuild something that is complex without it changing.
If unsure whether something is complicated or complex, then Foden’s test is to try replacing the adjective “complex” with the phrase “a complex system”. e.g. the market is a complex system (makes sense) vs the book is a complex system (doesn’t make sense).