It depends what you are talking about! 'Complex' implies a certain level of difficulty, either to create or to understand the pattern. 'Intricate' might refer to the appearance of the pattern.

It's not a good idea to choose phases based on "most widely used". This just limits your English progress. As Su.Ki. has written, "complex" suggests some difficulty. A complex pattern would not be easy to comprehend immediately. "Intricate" suggests precision and detail. These words aren't exactly the same.

The difference is subtle. "Complex" is a simpler and more everyday word and applies to more situations. To me, "complex" means having many different parts or elements, and, as a result being difficult to understand. To me, "intricate" suggests a really huge number of tiny and beautifully worked details. It's all opinion, but to me, a linkage like this https://momath.org/wp-content/uploads/positions.gif is "complex," because it's hard to understand how the parts move in relationship to each other, but it's not "intricate," because the parts are large and simple and I could build one out of cardboard in half an hour. On the other hand, the scrollwork on currency, like this: https://upload.wikimedia.org/wikipedia/commons/5/58/10_won_serieI_reverse.jpeg is "intricate" because I would need to use a magnifying glass to see the details. Many might say it's both "intricate" and "complex. On the other hand, I feel it isn't very "complex" because it isn't hard to understand--it's just a decorative frame."