I've tried to find the differences between 柄 and 模様 everywhere but to no avail.
What are the differences between 柄 and 模様?
柄 is closer to style, and it also means character (as in [人柄]{ひとがら}). 模様 is closer to objective pattern, and it also means appearance, current situation (as in [空模様]{そらもよう}).
When used for the style of clothes, skin of cats, etc., 柄 and 模様 are basically interchangeable. But I feel ~柄 is used mainly for organic patterns, and ~模様 is more common for inorganic/geometric/abstract patterns.
文様【もんよう】 is also used for traditional and complicated patterns such as Arabesque.
Only 模様 can be used with the patterns of rocks, electronic circuits, etc, because they're not related to decorating something/someone.