Intricate things are complex and have many elements: they're not simple. Think of the intricate wiring of a computer's motherboard, or the intricate plot of a movie that you have a hard time following.
I couldn't find any beauty things in here. It sounded more like complicated to me.
Intricate is more closely related to elaborate, while complicated is more closely related to complex.
Sometimes complicated and intricate can be used synonomously, but intricate has other meanings, especially as it relates to "elaborate details" which is viewed in a more favorable way.
For example, "Buildings of the Beaux-Arts architectural style feature many ornate and intricate