- To organize means to group things in a logical way so they are easy to find, manage, and work with.
One of the most commonly used examples is to organize your closet. It entails that your closet is messy and it's not easy for you to find your clothes, let alone access them. So, what you mean by organizing your closet is to group items logically: let's say you group pants, shirts, t-shirts, accessories, and so forth in different categories (things grouped together because they have something in common). And, place them anywhere in the closet.
- To arrange means to "place" things in a specific order to achieve a desired outcome: maybe for aesthetics, differentiation, optimizing space, accessibility, or anything.
For example: after making categories for your clothes might place your pants in your "pants" category according to weekdays: Brown pants on Monday, Grey pants on Tuesday, and so forth. And, you might arrange each item in each category according to weekdays. This arrangement is done to save you the time each morning you spend deciding what to wear for the office. And, last but not least, you might then arrange the categories according to your own preference. I like to arrange my clothes categories as such: First, shirts, then, t-shirts, trousers, denim, and joggers, and lastly, for accessories I have a different locker section in my closet. This arrangement indicated the order I wear my clothes: I wear a shirt, then pants, then accessories (I know it's weird but still makes sense to me).
P.S. These terms are really important to know. For context, an entire field is dedicated to organise data in computers: Data Structures.