They basically mean the same thing,
She wore the appropriate clothing for the event
She wore the proper clothing for the event
She wore clothing that was suitable for the event
1 difference is appropriate and proper can be used to describe behavior, while if you use suitable it might sound awkward.
One thing I realized in talk is that proper is used in more formal talks than the other two.
And to answer Hassan's question reasonable could be used the same as the others, but not always. Reasonable is usually used to describe something is close unlike the other words are used to describe something that is exactly right/wrong.
Ex: She wore reasonable clothing for the event
It sounds as thought she didn't dress exactly like she should have but she still dressed well.
I hope that helped, if you have any questions feel free to ask!