First of all, I think "intricate" has a positive connotation, and "complicated" has a negative connotation. They both essentially mean the same thing, but "intricate" sounds more positive. Some examples - "I developed an intricate system for studying English." "This Mercedes (car) is very intricate and well crafted." "Michaelangelo's painting in the Sistine Chapel is quite intricate."

intricate =difficult to understand; obscure; complex; puzzling *The math problem was very. intricate. On the other hand, complicated refers to a high level of difficulty. If a problem is complicated, there might be or might not be many parts but it will certainly take a lot of hard work to solve.